ALL p-LOCAL FINITE GROUPS OF RANK TWO FOR ODD PRIME p
ANTONIO D'IAZ, ALBERT RUIZ, AND ANTONIO VIRUEL
Abstract. In this paper we give a classification of the rank two p-local*
* finite groups for
odd p. This study requires the analisis of the possible saturated fusion*
* systems in terms of
the outer automorphism group ant the proper F-radical subgroups. Also, f*
*or each case in
the classification, either we give a finite group with the corresponding*
* fusion system or we
check that it corresponds to an exotic p-local finite group, getting som*
*e new examples of
these for p = 3.
1. Introduction
When studying the p-local homotopy theory of classifying spaces of finite gro*
*ups, Broto-
Levi-Oliver [11] introduced the concept of p-local finite group as a p-local an*
*alogue of the
classical concept of finite group. These purely algebraic objects, whose basic *
*properties are
reviewed in Section 2, are a generalization of the classical theory of finite g*
*roups in the sense
that every finite group leads to a p-local finite group, although there exist e*
*xotic p-local
finite groups which are not associated to any finite group as it can be read in*
* [11, Sect. 9],
[30], [35] or Theorem 5.9 below. Besides its own interest, the systematic stud*
*y of possible
p-local finite groups, i.e. possible p-local structures, is meaningful when wor*
*king in other
research areas as transformation groups (e.g. when constructing actions on sphe*
*res [2, 1]), or
modular representation theory (e.g. the study of the p-local structure of a gro*
*up is a very first
step when verifying conjectures like those of Alperin [3] or Dade [17]). It al*
*so provides an
opportunity to enlighten one of the highest mathematical achievements in the la*
*st decades:
The Classification of Finite Simple Groups [24]. A milestone in the proof of th*
*at classification
is the characterization of finite simple groups of 2-rank two (e.g. [22, Chapte*
*r 1]) what is based
in a deep understanding of the 2-fusion of finite simple groups of low 2-rank [*
*21, 1.35, 4.88].
Unfortunately, almost nothing seems to be known about the p-fusion of finite gr*
*oups of p-rank
two for an odd prime p [19], and this work intends to remedy that lack of infor*
*mation by
classifying all possible saturated fusion systems over finite p-groups of rank *
*two, p > 2.
Theorem 1.1. Let p be an odd prime and S be a rank two p-group. Given a saturat*
*ed fusion
system (S, F), one of the following hold:
o F has no proper F-centric, F-radical subgroups and it corresponds to the*
* group S :
Out F(S).
______________
Key words: 2000 Mathematics subject classification 55R35, 20D20.
First author is partially supported by MCED grant AP2001-2484.
First and third authors are partially supported by grant MCYT-BFM2001-1825.
Second author is partially supported by MCYT grant BFM2001-2035.
1
2 ANTONIO D'IAZ, ALBERT RUIZ, AND ANTONIO VIRUEL
o F is the fusion system of a group G which fits in the following extensio*
*n:
1 ! S0 ! G ! W ! 1
where S0 is a subgroup of index p in S and W contains SL2(p).
o F is the fusion system of an extension of one of the following finite si*
*mple groups:
- L3(p) for any p,
- 2F4(2)0, J4, L3(q), 3D4(q), 2F4(q) for p = 3, where q is a 30prime p*
*ower,
- T h for p = 3,
- He, F i024, O0N for p = 7 or
- M for p = 13.
o F is an exotic fusion system characterized by the following data:
- S = 71+2+and all the rank two elementary abelian subgroups are F-rad*
*ical,
- S = G(3, 2k; 0, fl, 0) and the only proper F-radical, F-centric subg*
*roups are one
or two S-conjugacy classes of rank two elementary abelian subgroups,
- S = G(3, 2k + 1; 0, 0, 0) and the proper F-centric, F-radical subgro*
*ups are ei-
ther one or two S-conjugacy classes of subgroups isomorphic to 31+2+*
*, either one
S-conjugacy class of rank two elementary abelian subgroups and one s*
*ubgroup
isomorphic to Z=3k + Z=3k.
Proof.For an odd prime p, the isomorphism type of a rank two p-group, namely S,*
* is described
in Theorem A.1, hence the proof is done by studying the different cases of S.
For p > 3, Theorems 4.1, 4.2 and 4.3 show that any saturated fusion system (S*
*, F) is
induced by S : OutF (S), i.e. S is resistant (see Definition 3.1), unless S ~=p*
*1+2+, hence [35]
completes the proof for the p > 3 case.
For p = 3, Theorems 4.1, 4.2, 5.1 and 5.8 describe all rank two 3-groups whic*
*h are resis-
tant. The saturated fusion system over non resistant rank two 3-groups are then*
* obtained in
Theorems 4.8 and 5.9 when S 6~= 31+2+, what completes the information in [35] a*
*nd finishes
the proof.
It is worth noticing that along the proof of the theorem above some interesti*
*ng contributions
are made:
o The Appendix provides a neat compendium of the group theoretical propert*
*ies of
rank two p-groups, p odd, including a description of their automorphism *
*groups,
and p-centric subgroups. It does not only collect the related results in*
* the literature
[6, 7, 18, 19, 26], but extends them.
o Theorems 4.1, 4.2, 4.3, 5.1 and 5.8 identify a large family of resistant*
* groups comple-
menting the results in [37], and extending those in [32].
o Theorem 5.9 provides infinite families of exotic rank two p-local finite*
* groups with
arbitrary large Sylow p-subgroup. Unlikely the other known infinite fami*
*lies of exotic
p-local finite groups [13, 30], some of these new families cannot be con*
*structed as
the homotopy fixed points of automorphism of a p-compact group. Neverthe*
*less, it is
still possible to construct an ä scending" chain of exotic p-local finit*
*e groups whose
colimit, we conjecture, should provide an example of an exotic p-local c*
*ompact group
[12, Section 6].
p-LOCAL FINITE GROUPS OF RANK TWO FOR ODD p *
* 3
Organization of the paper: Along Section 2 we quickly review the basics on p-lo*
*cal finite
groups. In Section 3 we define the concept of resistant p-group, similar to th*
*at of Swan
group, and develop some machinery to identify resistant groups. In Section 4 we*
* study the
fusion systems over non maximal nilpotency class rank two p-groups, while the s*
*tudy of fusion
systems over maximal nilpotency class rank two p-groups is done in Section 5. W*
*e finish this
paper with an Appendix collecting the group theoretical background on rank two *
*p-groups
which is needed along the classification.
Notation: By p we always denote an odd prime, and S a p-group of size pr. The *
*group
theoretical notation used along this paper is that described in the Atlas [15, *
*5.2]. For a
group G, and g 2 G, we denote by cg the conjugation morphism x 7! g-1xg. If P, *
*Q G, the
set of G-conjugation morphisms from P to Q is denoted by Hom G(P, Q), so if P =*
* Q then
Aut G(P ) = Hom G(P, P ). Notice that AutG (G) is then Inn(G), the group of inn*
*er automor-
phisms of G. Given P and Q groups, Inj(P, Q)denotes the set of injective homomo*
*rphisms.
Acknowledges: The authors are indebted with Ian Leary, Avinoam Mann and Bob Oli*
*ver.
The first two provided us the references for stating Theorem A.1, while the lat*
*er has shown
interest and made helpful comments and suggestions throughout the course of thi*
*s work.
2. p-local finite groups
At the beginning of this section we quickly review the concept of p-local fin*
*ite group
introduced in [11] that builds on a previous unpublished work of L. Puig, where*
* the axioms
for fusion systems are already established. See [12] for a survey on this subje*
*ct.
After that we study some particular cases where a saturated fusion system is *
*controlled by
the normalizer of the Sylow p-subgroup.
We end this section with some tools which allow us to study the exoticism of *
*an abstract
saturated fusion system.
Definition 2.1. A fusion system F over a finite p-group S is a category whose o*
*bjects are the
subgroups of S, and whose morphisms sets Hom F(P, Q) satisfy the following two *
*conditions:
(a) Hom S(P, Q) Hom F(P, Q) Inj(P, Q) for all P and Q subgroups of S.
(b) Every morphism in F factors as an isomorphism in F followed by an inclusion.
We say that two subgroups P ,Q S are F-conjugate if there is an isomorphism*
* between
them in F. As all the morphisms are injective by condition (b), we denote by Au*
*tF (P ) the
group Hom F(P, P ). We denote by OutF (P ) the quotient group AutF (P )= AutP(P*
* ).
The fusion systems that we consider are saturated, so we need the following d*
*efinitions:
Definition 2.2. Let F be a fusion system over a p-group S.
o A subgroup P S is fully centralized in F if |CS(P )| |CS(P 0)| for a*
*ll P 0which is
F-conjugate to P .
o A subgroup P S is fully normalized in F if |NS(P )| |NS(P 0)| for al*
*l P 0which is
F-conjugate to P .
o F is a saturated fusion system if the following two conditions hold:
(I)Every fully normalized subgroup P S is fully centralized and Aut *
*S(P ) 2
Sylp(Aut F(P )).
4 ANTONIO D'IAZ, ALBERT RUIZ, AND ANTONIO VIRUEL
(II)If P S and ' 2 Hom F(P, S) are such that 'P is fully centralized, *
*and if we set
N' = {g 2 NS(P ) | 'cg'-1 2 AutS('P )},
then there is __'2 Hom F(N', S) such that __'|P = '.
Remark 2.3. From the definition of fully normalized and the condition (I) of sa*
*turated
fusion system we get that if F is a saturated fusion system over a p-group S th*
*en p cannot
divide the order of the outer automorphism group OutF (S).
As expected, every finite group G gives rise to a saturated fusion system [11*
*, Proposi-
tion 1.3], which provides valuable information about BG^p[33]. Some classical *
*results for
finite groups can be generalized to saturated fusion systems, as for example, A*
*lperin's fusion
theorem for saturated fusion systems [11, Theorem A.10]:
Definition 2.4. Let F be any fusion system over a p-group S. A subgroup P S i*
*s:
o F-centric if P and all its F-conjugates contain their S-centralizers.
o F-radical if Out F(P ) is p-reduced, that is, if Out F(P ) has no nontri*
*vial normal p-
subgroups.
Theorem 2.5 (Alperin's fusion theorem for saturated fusion systems). Let F be a*
* saturated
fusion system over S. Then for each morphism _ 2 AutF (P, P 0), there exists a *
*sequence of
subgroups of S
P = P0, P1, . .,.Pk = P 0 and Q1, Q2, . .,.Qk,
and morphisms _i2 AutF (Qi), such that
o Qi is fully normalized in F, F-radical and F-centric for each i;
o Pi-1, Pi Qi and _i(Pi-1) = Pi for each i; and
o _ = _k O _k-1O . .O._1.
The subgroups Qi's in the theorem above determine the structure of F, so they*
* deserve a
name:
Definition 2.6. Let F be any fusion system over a p-group S. We say that a subg*
*roup Q S
is F-Alperin if it is fully normalized in F, F-radical and F-centric.
The definition of p-local finite group still requires one more new concept.
Let Fc denote the full subcategory of F whose objects are the F-centric subgr*
*oups of S.
Definition 2.7. Let F be a fusion system over the p-group S. A centric linking*
* system
associated to F is a category L whose objects are the F-centric subgroups of S,*
* together with
a functor
ß :L -! Fc
ffiP
and "distinguished" monomorphisms P -! Aut L(P ) for each F-centric subgroup P *
* S,
which satisfy the following conditions:
(A) ß is the identity on objects and surjective on morphisms. More precisely*
*, for each pair
of objects P, Q2L, Z(P ) acts freely on Mor L(P, Q) by composition (upon*
* identifying
Z(P ) with ffiP(Z(P )) AutL(P )), and ß induces a bijection
~=
MorL(P, Q)=Z(P )- ! Hom F(P, Q).
p-LOCAL FINITE GROUPS OF RANK TWO FOR ODD p *
* 5
(B) For each F-centric subgroup P S and each g 2 P , ß sends ffiP(g) 2 Aut*
* L(P ) to
cg 2 AutF (P ).
(C) For each f 2 Mor L(P, Q) and each g 2 P , the equality ffiQ(ß(f)(g)) O f*
* = f O ffiP(g)
holds in L
Finally, the definition of p-local finite group is:
Definition 2.8. A p-local finite group is a triple (S, F, L), where S is a p-gr*
*oup, F is a
saturated fusion system over S and L is a centric linking system associated to *
*F. The
classifying space of the p-local finite group (S, F, L) is the space |L|^p.
Given a fusion system F over the p-group S, there exists an obstruction theor*
*y for the
existence and uniqueness of a centric linking system, i.e. a p-local finite gr*
*oup, associated
to F. The question is solved for p-groups of small rank by the following result*
* [11, Theorem E]:
Theorem 2.9. Let F be any saturated fusion system over a p-group S. If rkp(S) <*
* p3, then
there exists a centric linking system associated to F. And if rkp(S) < p2, then*
* there exists a
unique centric linking system associated to F.
As all p-local finite groups studied in this work are over rank two p-groups *
*S, we obtain:
Corollary 2.10. Let p be an odd prime. Then the set of p-local finite groups ov*
*er a rank two
p-group S is in bijective correspondence with the set of saturated fusion syste*
*ms over S.
In [11, Section 2] is defined the "centralizer" fusion system of a given full*
*y centralized
subgroup:
Definition 2.11. Let F be a fusion system over S and P S a fully centralized *
*subgroup
in F. The centralizer fusion system of P in F, CF (P ) is the fusion system ove*
*r CS(P ) with
objects Q CS(P ) and morphisms,
Hom CF(P)(Q, Q0)
= {' 2 Hom F(Q, Q0)|9_ 2Hom F(QP, Q0P ), _|Q = ', _|P = IdP} .
Remark 2.12. If we consider the fusion system corresponding to a finite group G*
* with Sylow
p-subgroup S and P S such that CS(P ) 2 Sylp(CG(P )), i.e. P is fully central*
*ized in FS(G),
then the fusion system (CS(P ), CF (P )) is the fusion system (CS(P ), FCS(P)(C*
*G(P ))), so it is
again the fusion system of a finite group.
In [11, Section 6] the ön rmalizer" fusion system of a given fully normalized*
* subgroup is
defined:
Definition 2.13. Let F be a fusion system over S and P S a fully normalized s*
*ubgroup
in F. The normalizer fusion system of P in F, NF (P ) is the fusion system over*
* NS(P ) with
objects Q NS(P ) and morphisms,
Hom NF(P)(Q, Q0)
= {' 2 Hom F(Q, Q0)|9_ 2Hom F(QP, Q0P ), _|Q = ', _|P 2Aut (P )} .
Also in [11, Section 6] it is proved that if F is a saturated fusion system o*
*ver S and P is a
fully normalized subgroup then NF (P ) is a saturated fusion system over NS(P ).
6 ANTONIO D'IAZ, ALBERT RUIZ, AND ANTONIO VIRUEL
Remark 2.14. For a fusion system F over S, when considering the normalizer fusi*
*on system
over the own Sylow S, it turns out that NF (S) = F if and only if every ' 2 Hom*
* F(Q, Q0)
extends to _ 2 AutF (S) for each Q, Q0 S.
Moreover, we have the following two characterizations of F reducing to the no*
*rmalizer of
the Sylow:
Lemma 2.15. Let F be a saturated fusion system over the p-group S. Then F = NF *
*(S) if
and only if S itself is the only F-Alperin subgroup of S.
Proof.If S is the unique F-Alperin subgroup then the assertion follows from Alp*
*erin's fusion
theorem for saturated fusion systems (Theorem 2.5). Assume then that F = NF (S*
*) and
choose P S F-Alperin. Using that F = NF (S), it is straightforward that Out *
*S(P ) is
normal in OutF (P ) and, as P is F-radical, it must be trivial. Then P = NS(P )*
*, and as S is
a p-group, P must be equal to S.
Lemma 2.16. Let F be a saturated fusion system over the p-group S. Then F = NF *
*(S) if
and only if NF (P ) = NNF(P)(NS(P )) for every P S fully normalized in F.
Proof.Assume first that F = NF (S) and P S is fully normalized in F. In gener*
*al we have
that NF (P ) NNF(P)(NS(P )). The other inclusion follows if all ' 2 Hom NF(P)*
*(Q, Q0) which
extend to a morphism _ 2 Hom F(P Q, P Q0) such that _|Q = ' and _|P 2 AutF (P )*
*, extend
to an element of Aut(NS(P )). But using Remark 2.14 we get that _ extends to an*
* element
of AutF (S), which restricts to an element of AutF (NS(P )) because _ restricts*
* to an element
of Aut(P ).
Assume now that for every P S fully normalized in F we have NF (P ) = NNF(P*
*)(NS(P )).
According to Lemma 2.15 we have to check that S does not contain any proper F-A*
*lperin
subgroup. Let P be a F-Alperin subgroup, then it is NF (P )-Alperin too. But,*
* applying
Lemma 2.15 to NF (P ), we get that S = P .
Finally in this section we give some results which allow us to determine in s*
*ome special
cases the existence of a finite group with a fixed saturated fusion system.
We begin with a definition which does only depend on the p-group S:
Definition 2.17. Let S be a p-group. A subgroup P S is p-centric in S if CS(P*
* ) = Z(P ).
Remark 2.18. If F is any fusion system over the p-group S, then F-centric subgr*
*oups are
p-centric subgroups too.
The following result is a generalization of [11, Proposition 9.2] which apply*
* to some of our
cases. Recall that given a fusion system (S, F), a subgroup P S is called str*
*ongly closed
in F if no element of P if F-conjugate to any element of S \ P .
Proposition 2.19. Let (S, F) be a saturated fusion system such that every non t*
*rivial strongly
closed subgroup P S is non elementary abelian, p-centric and does not factori*
*ze as a product
of two or more subgroups which are permuted transitively by AutF (P ). Then if *
*F is the fusion
system of a finite group, it is the fusion system of a finite almost simple gro*
*up.
p-LOCAL FINITE GROUPS OF RANK TWO FOR ODD p *
* 7
Proof. Suppose F = FS(G) for a finite group G. Assume also that #|G| is minimal*
* with this
property. Consider H C G a minimal non trivial normal subgroup in G. Then H \*
* S is a
strongly closed subgroup of (S, F). If H \ S = 1 then F is also the fusion syst*
*em of G=H,
which contradicts the assumption of minimality on G. Now P def=H \S is a non tr*
*ivial normal
strongly closed subgroup in (S, F) and, as H is normal, P is the Sylow p-subgro*
*up in H. By
[20, Theorem 2.1.5] H must be either elementary abelian or a product of non abe*
*lian simple
groups isomorphic one to each other which must be permuted transitively by NG(H*
*) = G
(now by minimality of H). Notice that H cannot be elementary abelian as that wo*
*uld imply
that P is so while this is not possible by hypothesis. Therefore H is a product*
* of non abelian
simple groups. If H is not simple (so there is more than one factor) this would*
* break P into
two ore more factors which would be permuted transitively, so H must be simple.*
* Now, as P
is p-centric, CG(H) \ S P H, so CG(H) \ S CG(H) \ H = 1 and CG(H) = 1 (CG*
*(H)
is a normal subgroup in G of order prime to p, so if CG(H) 6= 1 taking G=CG(H) *
*gives again
a contradiction with the minimality of #|G|). This tells us that H C G Aut(H)*
*, so G is
almost simple.
Remark 2.20. In fact [11, Proposition 9.2] proves that if the only non trivial *
*strongly closed
p-subgroup in (S, F) is S, and moreover S is non abelian and it does not factor*
*ize as a product
of two or more subgroups which are permuted transitively by AutF (S), then if (*
*S, F) is the
fusion system of a finite group, it is the fusion system of an extension of a s*
*imple group by
outer automorphisms of order prime to p.
We finish this section with the following result, which can be found in [9, C*
*orollary 6.17]:
Lemma 2.21. Let (S, F) be a saturated fusion system, and assume there is a nont*
*rivial
subgroup A Z(S) which is central in F (i.e. CF (A) = F). Then F is the fusion*
* system of
a finite group if and only if F=A is so.
3. Resistant p-groups
In this section we recall the notion of Swan group and introduce its generali*
*zation for fusion
systems, as well as we discuss some related results. Some of these results wer*
*e considered
independently by Stancu in [36].
For a fixed prime p we recall that a subgroup H G is said to control (stron*
*g) p-fusion
in G if H contains a Sylow p-subgroup of G and whenever P , g-1P g H for P a *
*p-subgroup
of G then g = hc where h 2 H and c 2 CG(P ).
If we focus on a p-group S we may wonder if NG(S) controls p-fusion in G when*
*ever
S 2 Sylp(G). If this is always the case, then S is called a Swan group. The e*
*quivalent
concept in the setting of p-local finite groups is:
Definition 3.1. A p-group S is called resistant if whenever F is a saturated fu*
*sion system
over S then NF (S) = F.
Remark 3.2. Considering the saturated fusion system associated to S 2 Sylp(G) i*
*t is clear
that every resistant group is a Swan group. In the opposite way, up to date the*
*re is no known
Swan group that is not a resistant group.
8 ANTONIO D'IAZ, ALBERT RUIZ, AND ANTONIO VIRUEL
Following [32, Section 2] we look for conditions on a p-group S for being res*
*istant. If the
p-group S is resistant, when treating with a fusion system F over S all morphis*
*ms in F
are restrictions of F-automorphisms of S. In the general case, we must pay att*
*ention to
possible F-Alperin subgroups to understand the whole category F. The first ste*
*p towards
this objective is to examine p-centric subgroups.
Theorem 3.3. Let S be a p-group. If every proper p-centric subgroup P fi S veri*
*fies
OutS(P ) \ Op(Out (P )) 6= 1 ,
then S is a resistant group.
Proof.Let F be a saturated fusion system over S. According to Lemma 2.15, it is*
* enough to
prove that S is the only F-Alperin subgroup. Let P fi S be a proper F-Alperin s*
*ubgroup,
hence p-centric by Remark 2.18. As OutS(P ) OutF (P ), we have that
1 6= OutS(P ) \ Op(Out (P )) OutF (P ) \ Op(Out (P )) E Op(Out F(P )*
*) .
Hence P cannot be F-radical.
We obtain a family of resistant groups:
Corollary 3.4. Abelian p-groups are resistant groups.
Also is meaningful determining whether or not a p-group can be F-Alperin for *
*some satu-
rated fusion system F:
Lemma 3.5. Let P be a p-group such that Op(Out (P )) is the Sylow p-subgroup of*
* Out (P ).
Then P is not F-Alperin for any saturated fusion system F over S with P fi S.
Proof.Let S be a p-group with P fi S and F be a saturated fusion system over S.*
* If P is
F-radical then the normal p-subgroup Op(Out (P )) \ OutF (P ) of Out F(P ) must*
* be trivial.
Being Op(Out (P )) a Sylow p-subgroup and normal this implies that Out F(P ) is*
* a p0-group
and so, by definition, Aut P(P ) 2 Sylp(Aut F(P )). If in addition P is fully n*
*ormalized then
we know that AutS(P ) is another Sylow p-subgroup of AutF (P ). So they both mu*
*st be same
size. Finally, if P is F-centric then Z(P ) = CS(P ) and P = NS(P ), which is f*
*alse for p-groups
unless P is equal to S.
We obtain some useful corollaries:
Corollary 3.6. Z=pn is not F-Alperin in any saturated fusion system F over S wh*
*ere Z=pn fi
S.
Proof.An easy verification shows that Aut(Z=pn) is equal to (Z=pn)*, which is a*
*belian. Now
just apply Lemma 3.5.
Corollary 3.7. If n and m are two different non zero positive integers, then Z=*
*pn x Z=pm
is not F-Alperin in any saturated fusion system F over S where Z=pn x Z=pm fi S.
Proof.Suppose n < m and take f 2 End (Z=pn x Z=pm ) with f(~1, ~0) = (~a, ~b) a*
*nd f(~0, ~1) =
(~c, ~d). It must hold that b 0 mod pm-n . Moreover, f is an automorphism i*
*f and only if
order(~a, ~b) = pn, order(~c, ~d) = pm and <(~a, ~b), (~c, ~d)> = Z=pn x Z=pm .*
* It can be checked that
p-LOCAL FINITE GROUPS OF RANK TWO FOR ODD p *
* 9
the first condition is equivalent to a 6= 0 mod p, the second to d 6= 0 mod p, *
*and the third
is consequence of the previous ones. Counting elements it turns out that Aut(Z=*
*pn x Z=pm )
has order p3n+m-2(p - 1)2. Finally, the subgroup {(~a, ~b), (~c, ~d) with a = 1*
* mod p and d = 1
mod p} is normal and has order p3n+m-2.
Corollary 3.8. If p is odd then non abelian metacyclic p-groups M are not F-Alp*
*erin in any
saturated fusion system F over S with M fi S.
Proof. According to [18, Section 3], for p odd and M a non abelian metacyclic p*
*-group,
Op(Out (M)) is the Sylow p-subgroup of Out(M), so the result follows using Lemm*
*a 3.5.
Recall the notation in Theorem A.1 for the families of p-rank two p-groups an*
*d Theorem A.2
for the maximal nilpotency class 3-rank two 3-groups.
Corollary 3.9. If p is odd then G(p, r; ffl) cannot be F-Alperin in any saturat*
*ed fusion system
over S, with G(p, r; ffl) fi S.
Proof. By [19, Proposition 1.6] Op(Out (G(p, r; ffl))) is the Sylow p-subgroup *
*of Out(G(p, r; ffl)),
so applying Lemma 3.5 G(p, r; ffl) could not be F-Alperin.
Corollary 3.10. B(3, r; fi, fl, ffi) is not F-Alperin in any saturated fusion s*
*ystem F over S
where B(3, r; fi, fl, ffi) fi S.
Proof. From [6] we have that the Frattini subgroup of B(3, r; fi, fl, ffi), (B*
*(3, r; fi, fl, ffi)), is
= . Consider the Frattini map
B(3, r; fi, fl, ffi)) ! B(3, r; fi, fl, ffi)= (B(3, r; fi, fl, f*
*fi)) ' <_s, __s1>
and the induced map
æ : Out(G) ! Aut(B(3, r; fi, fl, ffi)= (B(3, r; fi, fl, ffi))) ' *
*GL 2(3)
whose kernel is a 3-group. As fl1 is characteristic in B(3, r; fi, fl, ffi), f*
*or every class of mor-
phisms ' in Out(G) it holds that æ(')(<__s1>) <__s1>, and so the image of æ i*
*s contained in the
lower triangular matrices.
The subgroup generated by 1011is normal and is the Sylow 3-subgroup of the l*
*ower trian-
gular matrices of GL 2(3). Its preimage by æ is normal in Out(G) and, as the ke*
*rnel of æ is a
3-group, it is the Sylow 3-subgroup of Out(G) too. To finish the proof apply Le*
*mma 3.5.
The following result related to Corollary 3.7 is needed in successive section*
*s, but before we
recall here [35, Lemma 4.1] in a clearer form:
Lemma 3.11. Let G be a p-reduced subgroup (that is, G has no nontrivial normal *
*p-subgroup)
of GL 2(p), p 3. If p divides the order of G then SL2(p) G.
Proof. If p divides the order of the group G, then there is an element of order*
* p. As G is
p-reduced it cannot be the only one, so we have a subgroup of GL 2(p) with more*
* than one
nontrivial p-subgroup. Observe that the only nontrivial p-subroups in GL 2(p) a*
*re the Sylow
p-subgroups, so G has more than one Sylow p-subgroup. Using the Third Sylow Th*
*eorem
there are at least p + 1 different Sylow p-subgroups in G GL 2(p), but there *
*are exactly
p + 1 Sylow p-subgroups in GL 2(p), so G contains all the Sylow p-subgroups in *
*GL 2(p) and
the subgroup they generate, thus SL2(p) G.
10 ANTONIO D'IAZ, ALBERT RUIZ, AND ANTONIO VIRUEL
Proposition 3.12. For a prime p > 3 and an integer n > 1, Z=pn x Z=pn is not F-*
*Alperin
in any saturated fusion system F over a p-group S where Z=pn x Z=pn fi S.
Proof.Consider S a p-group and Z=pn x Z=pn fi S. If we assume that Z=pn x Z=pn *
*is F-
radical then G def=AutF(Z=pn x Z=pn) must be p-reduced, and if Z=pn x Z=pn is F*
*-centric it
is self-centralizing in S, so taking the conjugation by an element in S \ Z=pn *
*x Z=pn we get
that there exist an element of order p in G.
We can consider Aut(Z=pn x Z=pn) as 2 x 2 matrices with coefficients in Z=pn *
*and with
determinant non divisible by p. In that case the reduction modulo p induces a s*
*hort exact
sequence:
j
{1} ! P ! Aut(Z=pn x Z=pn) ! GL 2(p) ! {1} ,
with P a p-group. If the intersection P \ G is not trivial, then we have a non *
*trivial normal
p-subgroup in G and G would not be p-reduced, so G\P = {1} and æ restricts to a*
*n injection
of G in GL 2(p).
Using that æ(G) has an element of order p, and it is a p-reduced subgroup of *
*GL 2(p) and
applying the Lemma 3.11 we get that æ(G) contains SL2(p). In particular we get*
* that the
__def
matrix A = 1011is in_æ(G),_so we have an element in A 2 G Aut(Z=pn x Z=pn) *
*which
reduction modulo p is A . A must be matrix of the form A = 1+,p1+''p~p1+~p, *
*and an easy
computation tell us that
` m m m m '
1+m,p+ ~p m+ ,p+mjp+ ~p+ ~p 2
Am 2 2 m 3 2 mod p .
m~p 1+ 2 ~p+m~p
So, for p > 3, we get Ap 10p1 mod p2 and as n > 1, the order of A is bigger*
* than p,
which contradicts the fact that æ is injective.
If p equals 3 or n equals 1 then the thesis of the previous lemma is false, t*
*hat is, Z=pnxZ=pn
could be F-Alperin when p = 3 or n = 1. But then we can sharp our result in the*
* following
way:
Lemma 3.13. If p = 3 or n = 1 and if Z=pn x Z=pn is F-Alperin in a p-local fini*
*te group
(S, F, L) with Z=pn x Z=pn fi S, then p1+2+ S.
Proof.Let F be a saturated fusion system over S, and suppose that P def=Z=pn x *
*Z=pn fi S.
As in the proof of Proposition 3.12, if P is F-radical and F-centric then AutF *
*(P ) contains
SL 2(p).
Let ß :L -! Fc be the centric linking system, and take the short exact sequen*
*ce of groups
induced by ß:
1 ! P ! AutL(P ) i!Aut F(P ) ! 1.
Because AutF (P ) contains the special matrices over Fp we have another short e*
*xact sequence:
1 ! P ! M !i SL2(p) ! 1.
To see that there are no more extensions that the split one, consider the centr*
*al subgroup V
of SL 2(p) generated by the involution -100-1. Since p 3, multiplication by*
* |V | = 2 is
invertible in P and so Hk(V, P ) = 0 for all k > 0, and because V acts on P wit*
*hout fixed
p-LOCAL FINITE GROUPS OF RANK TWO FOR ODD p *
* 11
points also H0(V, P ) = 0. Now the Hochschild-Serre spectral sequence correspon*
*ding to the
normal subgroup V SL2(p) give us that Hk(SL 2(p), P ) = 0 for all k 0.
Now, as H2(SL 2(p), P ) = 0, the middle term M of the short exact sequence ab*
*ove equals
P :SL 2(p) AutL(P ).
As we are assuming in addition that the abelian p-group P is fully normalized*
* then, as F is
saturated, we obtain from Definitions 2.2 and 2.7(A) that the Sylow p-subgroup *
*of AutL(P )
is NS(P ). So then we have that p1+2+ P : Z=p NS(P ) S.
Lemma 3.11 also applies for giving some restrictions to the family C(p, r):
Lemma 3.14. Let F be a saturated fusion system over a p-group S with p 3. If *
*C(p, r) fi S
(with r 3) is F-centric and F-radical, then SL2(p) OutF (C(p, r)) GL 2(p).
Proof. If C(p, r) is F-radical, then OutF (C(p, r)) must be p-reduced. If we co*
*nsider the pro-
jection æ from Lemma A.5 we have that it must restrict to a monomorphism in Out*
*F (C(p, r))
(otherwise we would have a non-trivial normal p-subgroup) so we can *
*consider
Out F(C(p, r)) GL 2(p).
Now as C(p, r) is F-centric and different from S, we have an element of order*
* p in
Out F(C(p, r)). Now, using again that Out F(C(p, r)) must be p reduced, the re*
*sult follows
from Lemma 3.11.
4. Non-maximal class rank two p-groups
In this section we consider the non-maximal class rank two p-groups for odd p*
*, which are
listed in the classification in Theorem A.1.
We begin with metacyclic groups:
Theorem 4.1. Metacyclic p-groups are resistant for p 3.
Proof. We prove that if S is a metacyclic group then it cannot contain any prop*
*er F-Alperin
subgroup.
Let P be a proper subgroup of S, then it must be again metacyclic.
If P is not abelian, we can use Corollary 3.8 to deduce that it cannot be F-A*
*lperin.
If P is abelian, using Corollaries 3.6 and 3.7 it cannot be F-Alperin unless *
*P ~=Z=pnxZ=pn,
but in this case p1+2+should be a subgroup of M(p, r) by Lemma 3.13, which is i*
*mpossible
because p1+2+is not metacyclic.
In the study of C(p, r) in this section we assume r 4: for r = 3 we have th*
*at C(p, 3)~=p1+2+,
which it is a maximal nilpotency class p-group and the fusion systems over that*
* group are
studied in [35].
Theorem 4.2. If r > 3 and p 3 then C(p, r) is resistant.
Proof. Let F be a saturated fusion system over C(p, r). The possible proper F-*
*Alperin
subgroups are proper p-centric subgroups, and using Lemma A.6 these are isomorp*
*hic to
Z=pr-2x Z=p. But as r > 3, Z=pr-2x Z=p cannot be F-radical by Corollary 3.7.
It remains to study the non-maximal nilpotency class groups of type G(p, r; f*
*fl). Remark A.3
tells us that in this section we have to consider all of them but G(3, 4; 1).
12 ANTONIO D'IAZ, ALBERT RUIZ, AND ANTONIO VIRUEL
The study of groups of type G(p, r; ffl) is divided in two cases: for p 5 t*
*hese are resistant
groups, while for p = 3 we obtain saturated fusion systems with proper F-Alperi*
*n subgroups.
Theorem 4.3. If p > 3 and r 4, G(p, r; ffl) is resistant.
The proof needs the following lemma:
Lemma 4.4. Let F be a saturated fusion system over G(p, r; ffl) with p > 3 and *
*r 4. Then
C(p, r - 1) < G(p, r; ffl) is not F-radical. Moreover, Aut F(C(p, r - 1)) is a *
*subgroup of the
lower triangular matrices with first diagonal entry 1.
Proof.Consider p 5 and assume that C(p, r-1) is F-radical, then by Lemma 3.14*
* SL2(p)
Out F(C(p, r - 1)) and therefore the matrix x00x-1, where x is a primitive (p *
*- 1)-th root of
the unity in Fp, is the image of some ' 2 AutF (C(p, r - 1)) by the composition
j
AutF(C(p, r - 1))_i_//_OutF(C(p, r - 1))__//_GL2(p)
of the projection and æ from Lemma A.5. By Definition 2.2 this morphism ' must*
* lift
to Aut F(G(p, r; ffl)) as a morphism that maps a to ax (up to bncmp multiplicat*
*ion). But
automorphisms of G(p, r; ffl) map a to a 1 (up to bncmp multiplication), and -1*
* is not a
primitive (p - 1)-th root of the unity in Fp for p 5. Therefore C(p, r - 1) i*
*s not F-radical.
Finally, as C(p, r - 1) is not F-radical, then ß(Aut F(C(p, r - 1))) < NGL2(p*
*)(V ) where V
is the group generated by -1101, that is, the projection ß(Aut F(C(p, r - 1)))*
* is a subgroup
of the lower triangular matrices. Then all morphisms in Aut F(C(p, r - 1)) mus*
*t lift to
Aut F(G(p, r; ffl)), and again only those with 1 in the first diagonal entry a*
*re allowed, what
proves the second part of the lemma.
Proof of Theorem 4.3.If r > 4 we have just to consider the case of C(p, r-1) be*
*ing F-radical,
and Lemma 4.4 shows that it cannot be.
If r = 4 it remains to check what happens with the rank two elementary abelia*
*n subgroups
in G(p, 4; ffl). According to Lemma A.8 there are exactly p of these subgroups,*
* namely, Videf=
for i = 0, ..., p - 1, and all of them lie inside C(p, 3) ~=p1+2+. No*
*tice that conjugation
by c permutes all of them cyclically and so they are F-conjugate in any saturat*
*ed fusion
system over G(p, 4; "). Thus if any of them is F-radical then all of them are F*
*-radical.
If this is the case, fix x a primitive (p-1)-th root of the unity in Fp and l*
*et Vibe one of these
rank two p-subgroups with Fp basis . Then SL 2(p) Aut F(Vi) by Le*
*mma 3.11.
Now, the element x00x-12 AutF (V ) must lift to an automorphism of p1+2+by Def*
*inition 2.2.
The image of this extension in Out F(p1+2+) = GL 2(p) is a matrix Li with x as *
*eigenvalue
and with determinant x-1, so it has x-2 as the other eigenvalue. Notice that ea*
*ch Vi gives
a different matrix Li 2 GL 2(p), so different elements of order p - 1 in OutF (*
*p1+2+). But the
description of AutF (p1+2+) in Lemma 4.4 shows us that there are not such matri*
*ces Liif p > 5
and at most one when p = 5 (cf. [35, Section 4]).
So it remains to check the cases with p = 3, and as this section only copes w*
*ith the non
maximal nilpotency class groups, we consider r 5.
p-LOCAL FINITE GROUPS OF RANK TWO FOR ODD p *
* 13
Lemma 4.5. Fix r 4 and let W be either SL2(3) or GL 2(3). Given a faithful re*
*presentation
of Z=3 in Out(C(p, r - 1)) and any group S fitting in the exact sequence
1____//_C(3, r -_1)_//_S___//_Z=3___//_1
that induces the given representation of Z=3, there is a finite group G which f*
*its in the following
commutative diagram
1 ____//_C(3, r -_1)_//_S___//_Z=3___//_1
|| || |
|| | |
|| fflffl| fflffl|
1 ____//_C(3, r -_1)_//_G____//W ____//_1
where the last column is an inclusion of the Sylow 3-subgroup in W .
Proof. Fix a faithful representation of Z=3 in Out(C(p, r - 1)).
Consider first r = 4 or W = GL 2(3). Using the study of the group Aut (C(3, *
*r - 1)) in
Lemma A.5 we have that there is only one faithful representation of W in Out(C(*
*3, r - 1))
up to conjugation.
To compute the possible equivalence class of extensions in the exact sequence*
* 1 ! C(3, r -
1) ! G ! W ! 1 with the fixed representation of W in Out(C(3, r -1)) we have to*
* compute
H2(W ; Z(C(3, r - 1))) [8, Theorem IV.6.6], getting H2(W ; Z=3r-3) ~=Z=3.
Consider now Z=3, the Sylow 3-subgroup of W , and consider the following diag*
*ram:
1 ____//_C(3, r -_1)_//_S___//_Z=3___//_1
|| || |
|| | |
|| fflffl| fflffl|
1 ____//_C(3, r -_1)_//_G____//W ____//_1
with exact rows and do the same computation for the first row in cohomology. T*
*his gives
us that there are three possible equivalence classes of extensions S fitting in*
* that exact
sequence. Moreover the map induced in cohomology H2(W ; Z=3r-3) ! H2(Z=3; Z=3r-*
*3) is
an isomorphism (use a transfer argument), so we have that all three S appear as*
* the Sylow
3-subgroup of G.
Consider now the case r 5 and W = SL2(3). Now by Lemma A.5 we have two poss*
*ible
faithful representations of W in Out (C(p, r - 1)) up to conjugation. Now we d*
*o the same
computations and we use the same argument as before for each action, getting 6 *
*possible
equivalence classes of extensions which appear as the Sylow 3-subgroup of the 6*
* possible
G's.
Remark 4.6. The previous assertion fails for p > 3 because in the cohomology gr*
*oups
computation we get that H2(SL 2(p); Z(C(p, r-1))) and H2(GL 2(p); Z(C(p, r-1)))*
* are trivial,
so the only extensions are the split ones. So in both cases the Sylow p-subgrou*
*p has p-rank
three.
Remark 4.7. We are interested in the possible extensions 1 ! C(3, r - 1) ! S ! *
*Z=3 ! 1
such that the 3-rank of S is two. Looking at the groups appearing in the class*
*ification in
Theorem A.1 we get that the only possible ones are C(3, r), G(3, r; 1) and G(3,*
* r; -1).
14 ANTONIO D'IAZ, ALBERT RUIZ, AND ANTONIO VIRUEL
Fix the notation C(3, r - 1) .fflW for the groups in Lemma 4.5 with Sylow 3-s*
*ubgroup
isomorphic to G(3, r; ffl).
Theorem 4.8. The 3-local finite groups over G(3, r; ffl) with r 5 and at leas*
*t one proper
F-Alperin subgroup are classified by the following parameters:
______________________________________________________
| Out F(G(3, r; ffl))O|utF(C(3, r - 1)) |Group |
|_____________________|_________________|_____________|_
| Z=2 | SL 2(3) |C(3, r - 1) .fflSL2(3) |
|_________________|________________|__________________ |
| Z=2 x Z=2 | GL 2(3) |C(3, r - 1) .fflGL2(3) |
|_________________|________________|__________________ |
Table 1. s.f.s. over G(3, r; ffl) for r 5.
where the second column gives the outer automorphism group over the only proper*
* F-Alperin
subgroup, which is isomorphic to C(3, r-1). In the last column we refer to non *
*split extensions
such that the Sylow 3-subgroup is isomorphic to G(3, r; ffl) for ffl = 1 and i*
*t induces the desired
fusion system.
Proof.Assume now that C(3, r - 1) G(3, r; ffl) is F-Alperin. Then Out F(C(3,*
* r - 1)) ~=
SL 2(3) or OutF (C(3, r - 1)) ~=GL 2(3).
If we look at the possible Out F(G(3, r; ffl)) we get that it is Z=2, generat*
*ed by a matrix
which induces - Idin the identification OutF (C(3, r - 1)) GL 2(3) or Z=2 x Z*
*=2, generated
by - Idand a matrix of determinant -1 in OutF (C(3, r-1)) GL 2(3). From here *
*we deduce
that if OutF (G(3, r; ffl)) ~=Z=2 and C(3, r - 1) is F-radical then OutF (C(3, *
*r - 1)) ~=SL2(3),
while if OutF (G(3, r; ffl)) ~=Z=2 x Z=2 then OutF (C(3, r - 1)) ~=GL 2(3) what*
* completes the
table.
All of them are saturated because are the fusion systems of the finite groups*
* described in
Lemma 4.5.
5.Maximal nilpotency class rank two p-groups
In this section we classify the p-local finite groups over p-groups of maxima*
*l nilpotency
class and p-rank two. Recall that by Corollary 2.10 we have just to classify t*
*he saturated
fusion systems over these groups.
Consider S a p-rank two maximal nilpotency class p-group of order pr. For r *
*= 2 then
S ~=Z=p x Z=p, which is resistant by Corollary 3.4. If r = 3 then S ~=p1+2+and *
*this case has
been studied in [35]. For r 4 all the p-rank two maximal nilpotency class gro*
*ups appear
only at p = 3, and we use the description and properties given in Appendix A.
The description of the maximal nilpotency class 3-groups of order bigger than*
* 33 depends
on three parameters fi, fl and ffi, and we use the notation given in Theorem A.*
*2, so we call
the groups B(3, r; fi, fl, ffi) and {s, s1, s2, . .,.sr-1} the generators.
First we consider the non split case, that is, ffi > 0.
Theorem 5.1. Every group of type B(3, r; fi, fl, ffi), ffi > 0, is resistant.
p-LOCAL FINITE GROUPS OF RANK TWO FOR ODD p *
* 15
Proof. Let F be a saturated fusion system over B(3, r; fi, fl, ffi). Using Alpe*
*rin's fusion theorem
for saturated fusion systems (Theorem 2.5) it is enough to see whether B(3, r; *
*fi, fl, ffi) is the
only F-Alperin subgroup. So let P be a proper subgroup.
If P has 3-rank one then, it is cyclic and we can apply Corollary 3.6 to obta*
*in that P
cannot be F-Alperin.
Now assume that P has 3-rank two. Then, by Theorem A.1, P is one of the follo*
*wing:
o M(3, r) non abelian: according to Corollary 3.8 P can not be F-Alperin.
o G(3, r; ffl) group: it cannot be F-Alperin by Corollary 3.9.
o B(3, m; fi, fl, ffi) with m < r: by Lemma 3.10 P cannot be F-Alperin.
o C(3, r) group: as 31+2+= C(3, 3) is contained in C(3, r) we obtain that*
* 31+2+
B(3, r; fi, fl, ffi).
o Abelian: say P = Z=3m x Z=3n. If m 6= n then P cannot be F-Alperin by C*
*orol-
lary 3.7, and if m = n then again 31+2+ B(3, r; fi, fl, ffi) by Lemma 3*
*.13.
So if P is F-Alperin then B(3, r; fi, fl, ffi) must contain 31+2+.
We finish the proof by showing that 31+2+ B(3, r; fi, fl, ffi) since we are *
*in the non split case
(ffi > 0). Consider the short exact sequence:
1 ! fl1 ! B(3, r; fi, fl, ffi) i!Z=3 ! 1.
If ß(31+2+) is trivial then 31+2+ fl1. But by [26, Satz III.x14.17] fl1 is met*
*acyclic, and conse-
quently all its subgroups are metacyclic too. Thus fl1 cannot contain the C(3, *
*3) group. We
obtain then the short exact sequence:
1 ! Z=3 x Z=3 ! 31+2+i!Z=3 ! 1.
As this sequence splits, the same would holds for the exact sequence involving
B(3, r; fi, fl, ffi), and this is not the case.
We now consider the split case, that is ffi = 0. We prove that for fi = 1 th*
*ese maximal
nilpotency class groups are resistant, while for fi = 0 they are not. We use th*
*e information
about B(3, r; fi, fl, 0) contained in the appendix. According to Alperin's fusi*
*on theorem for
saturated fusion systems (Theorem 2.5) we must focus on the F-Alperin subgroups*
*. The next
two lemmas list the subgroups candidates for being F-Alperin in a saturated fus*
*ion system
over B(3, r; 0, fl, 0) and B(3, r; 1, 0, 0):
Lemma 5.2. Let F be a saturated fusion system over B(3, r; 0, fl, 0) and let P *
*be a proper
F-Alperin subgroup._Then_P_is_one_of_the_following_table:_______________
| Isomorphism |Subgroup (up to| |
| | |Conditions |
| type |conjugation) | |
|_____________|_______________|_______________________________ |
| Z=3k x Z=3k |fl1 = |r = 2k + 1. |
|_____________|_______________|______________________________||||
| 1+2 | def |i =s 3k-1, i0=s 3k-1for r =2k +|1,
| 3+ |Ei= *| 2 k-1 1 k-2 |
| | |i =s 3 , i0=s -3 for r =2k,|
|_____________|________________| 1 2 |
| | def |i 2 {-1, 0, 1} if fl = 0 and |
| Z=3 x Z=3 |V = ** | |
| | i 1 |i = 0 if fl = 1, 2. |
|_____________|_______________|______________________________|
Moreover, for the subgroups in the table, fl1 and any Ei are always F-centric, *
*while any Vi is
F-centric only if it is not F-conjugate to ** ~=Z=3 x Z=3.
16 ANTONIO D'IAZ, ALBERT RUIZ, AND ANTONIO VIRUEL
Proof.If P is F-Alperin then arguing as in Theorem 5.1 we obtain that P ~=Z=3n *
*x Z=3n or
P ~=C(3, n). Then, using Lemma A.15 we reach the subgroups in the statement.
To check that indeed fl1 is F-centric it is enough to notice that it is self-*
*centralizing and
that fl1 is F-conjugate just to itself in any fusion system F (fl1 is strongly *
*characteristic in
B(3, r; 0, fl, 0)). It is clear that the copies of 31+2+are F-centric because *
*these are the only
copies lying in B(3, r; 0, fl, 0), and the center of all of them is **. To con*
*clude the lemma,
notice that the only copies of Z=3 x Z=3 in B(3, r; 0, fl, 0) are the Vi's and *
***, and that
the former are self-centralizing while the centralizer of the latter is fl1.
Lemma 5.3. Let F be a saturated fusion system over B(3, r; 1, 0, 0) and P be a *
*proper F-
Alperin subgroup._Then_P_is_one_of_the_following_table:___________________
| Isomorphism |Subgroup (up | |
| | | |
| type |to conjuga-|Conditions |
| | | |
| |tion) | |
|________________|______________|__________________________________ |
| k-1 k-1 | | |
| Z=3 xZ=3 |fl2 = |r = 2k. |
|________________|______________|_________________________________||||
| 1+2 | def 0 | |
| 3+ |E0 = **| 3k-1 0 -3k-2 |
| | |i = s2 , i = s3 for r = 2k+1,|
|________________|______________| 3k-2 0 3k-2 |
| | def |i = s3 , i = s2 for r = 2k. |
| Z=3 x Z=3 |V0 = ** | |
|________________|______________|_________________________________|
Moreover, for the subgroups in the table, fl2 and E0 are always F-centric, and *
*V0 is F-centric
only if it is not F-conjugate to ** ~=Z=3 x Z=3.
Proof.The reasoning is totally analogous to that of the previous lemma using Le*
*mma A.16.
Remark 5.4. The isomorphism between 31+2+and Ei is given by a 7! i0, b 7! ss1i *
*and
c 7! i, where a, b and c are the generators of 31+2+= C(3, 3) given in Theorem *
*A.1. Therefore
the morphisms in Out (Ei) are described as in [35, Lemma 3.1] by means of the m*
*entioned
isomorphism, and for Out(Vi) we choose the ordered basis {i, ss1i}.
When studying a saturated fusion system F over B(3, r; 0, fl, 0) or B(3, r; 1*
*, 0, 0) it is enough
to study the representatives subgroups given in the tables of Lemmas A.15 and A*
*.16 because
F-properties are invariant under conjugation.
As a last step before the classification itself, we work out some information*
* on lifts and
restrictions of automorphism of some subgroups of B(3, r; fi, fl, 0). Given a s*
*aturated fusion
system F over B(3, r; fi, fl, 0) subsequently it will be advantageous to consid*
*er for every sub-
j
group of rank two P B(3, r; fi, fl, 0) the Frattini map Out(P ) ! GL2(3), wh*
*ich kernel is a
3 group, and its restriction
j
Out F(P ) ! GL 2(3) .
Notice that by Remark 2.3 this restriction is a monomorphism for P = B(3, r; fi*
*, fl, 0). For
P = fl1 or fl2 it is a monomorphism if P is F-Alperin as OutF (P ) is 3-reduced*
*. For P = Ei,
Vi they are inclusions by [35, Lemma 3.1] and by definition respectively. In t*
*he case this
restriction is a monomorphism we identify OutF (P ) with its image in GL 2(3) w*
*ithout explicit
mention. We divide the results in three lemmas, the first gives a description o*
*f OutF (P ) for P
an F-Alperin subgroup, the second copes with restrictions and the third with li*
*fts.
p-LOCAL FINITE GROUPS OF RANK TWO FOR ODD p *
* 17
Lemma 5.5. Let F be a saturated fusion system over B(3, r; fi, fl, 0). Then:
(a) æ(Out F(B(3, r; fi, fl, 0))) is a subgroup of the lower triangular matrices,
(b) Out F(B(3, 2k; 0, fl, 0)) Z=2 x Z=2,
(c) Out F(B(3, 2k + 1; 0, 1, 0)) Z=2,
(d) Out F(B(3, r; 1, 0, 0)) Z=2 and
(e) if P = fl1, fl2, Ei or Vi is F-Alperin then OutF (P ) = SL2(3) or GL 2(3).
Proof. For the first claim notice that from the order of Aut(B(3, r; fi, fl, 0)*
*) we deduce that
a 30element ' should have order 2 or 4. Now, as the Frattini subgroup of B(3, r*
*; fi, fl, 0) is
, the projection on the Frattini quotient becomes
j
OutF(B(3, r; fi, fl, 0)) ! GL 2(3)
0 e00 f0 f00 __ e0
where if __'maps s to ses1es2 and s1 to s1 s2 then æ(' ) = e0f0 . So we obt*
*ain lower
triangular matrices. Checking cases leads to obtain that the order of ' is two*
*, and that
' 2 { 100-1, -1001, -100-1, 110-1, -1011, -110-1, -10-11}. For fi = 0, f*
*l = 1 and odd r
or fi = 1 Lemma A.14 tells us that e must be equal to 1, and then it is easily *
*deduced that
Out F(B(3, r; fi, fl, 0)) must have order two.
For the last point just use Lemma 3.11 and that [GL 2(3) : SL2(3)] = 2.
Now we focus on the restrictions. For the study of the possible saturated fus*
*ion systems F
it is enough to consider diagonal matrices of OutF (B(3, r; fi, fl, 0)) instead*
* of lower triangular
ones. This is so because every Z=2 and Z=2 x Z=2 in OutF (B(3, r; fi, fl, 0)) i*
*s B(3, r; fi, fl, 0)-
conjugate to a diagonal one.
Lemma 5.6. Let F be a saturated fusion system over B(3, r; fi, fl, 0). Then:
(a) If fi = 0 the restrictions of the elements __'2 OutF (B(3, r; 0, fl, 0)) to*
* OutF (fl1) are given
by the following table, where it is also described the permutation of the E*
*i's induced
by __', and the restrictions to Out F(Ei0) for i0 2 {-1, 0, 1} such that Ei*
*0is fixed by the
permutation.
___________________________________________________
| 10| | -1 0 | -1 0 |
| Out F(B(3, r; 0, fl, 0))0|-1 | 0 1 | 0 -1 |
|__________________________|___|__________|________|
| |-1 0 | 1 0 | -10 |
| OutF(fl1) | 0-1 | 0 -1 | 01 |
|______________________|_______|__________|________|
| F-conjugation |E1 $ E-1 |E1 $ E-1 | - |
|____________________|_________|_________|_________|
| | -10 | 1 0 | -1 0 |
| Out F(Ei0), r odd | 0 1 | 0 -1 | 0 -1 |
|______________________|_______|__________|________|
| | -10 | -1 0 | 1 0 |
| OutF(Ei0), r even | 0 1 | 0 -1 | 0-1 |
|_____________________|________|__________|________|
(b) If fi = 1 (thus fl = 0) the restrictions to Out F(fl2) and to Out F(E0) of *
*outer automor-
phisms __'2 OutF (B(3, r; 1, 0, 0)) are given by the following table.
______________________________
| 1 0| |
| Out F(B(3, r; 1, 0, 0))0|-1|
|_________________________|__|_
| |-10 |
| Out F(fl2) |0-1 |
|______________________|_____|_
| |-1 0 |
| Out F(E0) | 0 1 |
|_____________________|______|_
18 ANTONIO D'IAZ, ALBERT RUIZ, AND ANTONIO VIRUEL
(c) For the outer automorphism groups of Ei and Vi we have the following restri*
*ctions:
____________________________________
| | 10 | -10 | -1 0 |
| Out F(Ei) | 0-1 | 0 1 | 0 -1 |
|___________|______|_______|________|
| |-1 0 | -10 | 1 0 |
| OutF(Vi) | 0-1 | 0 1 | 0-1 |
|___________|______|_______|________|
Proof.These are the only possible elements by Lemma 5.5, and the restrictions a*
*re computed
directly using the explicit form of the subgroups given in Lemmas 5.2 and 5.3 u*
*sing the basis
described in Remark 5.4.
Finally we reach the lemma about lifts:
Lemma 5.7. Let F be a saturated fusion system over B(3, r; fi, fl, 0), and P be*
* one of the
proper subgroups appearing in the tables of Lemmas 5.2 or 5.3. Then every diag*
*onal outer
automorphism of P in F (like those appearing in Lemma 5.6) can be lifted to th*
*e whole
B(3, r; fi, fl, 0). In particular, every admissible diagonal outer automorphis*
*m of P in F is
listed in the tables of restrictions of Lemma 5.6.
Proof.We study the two cases fi = 0 and fi = 1 separately.
If fi = 0 we begin with the morphisms from B(3, r; 0, fl, 0) restricted to fl*
*1. If fl1 is not
F-Alperin then every morphism in Aut F(fl1) can be lifted by Theorem 2.5. Supp*
*ose then
that fl1 is F-Alperin. Take ' 2 OutF (fl1) appearing in the table of Lemma 5.6 *
*and consider
the images cs, cs2 2 Out F(fl1) of the restrictions to fl1 of conjugation by s *
*and s2. Then it
can be checked that 'cs'-1 equals cs or cs2. Now apply (II) from Definition 2.2.
For Ei B(3, r; 0, fl, 0) there is a little bit more work to do. If Ei is not*
* F-Alperin apply
Theorem 2.5 again. So suppose Ei is F-Alperin, take __'2 OutF (Ei) and compute *
*N' from
Definition 2.2. The normalizer of Ei is ** ~=(Z=9 x Z=3) : Z=3 where *
*the order of i00
equals 9, and 'c``00'-1 equals c``00or c``002. So ' can be lifted to the normal*
*izer (Definition 2.2)
and, as this subgroup cannot be F-Alperin by Lemma 5.2, we can extend again to *
*the whole
B(3, r; 0, fl, 0) by Theorem 2.5. The last case is to consider Vi Ei. The deta*
*ils are analogous
with c``0.
If fi = 1, then P can be identified with a centric subgroup of B(3, r - 1;*
* 0, 0, 0) <
B(3, r; 1, 0, 0) (see proof of Lemma A.16) and then use the arguments above to *
*extend
the morphisms to B(3, r - 1; 0, 0, 0). We then use saturateness to extend the *
*morphism
to B(3, r; 1, 0, 0).
Theorem 5.8. Every rank two 3-group isomorphic to B(3, r; 1, 0, 0) is resistant.
Proof.Using Lemma 5.3, the only possible F-Alperin proper subgroups of B(3, r; *
*1, 0, 0)
are E0 and V0 if r is odd and fl2, E0 and V0 if r is even. In both cases, acco*
*rding to
Lemmas 5.7 and 5.6, if E0 (respectively V0) is F-Alperin, then OutF (E0) contai*
*ns the diagonal
matrix - Id(respectively the matrix 100-1) which is not admissible by Lemma 5.*
*7. Therefore
we are left with the case of r = 2k when fl2 is the only F-Alperin proper subgr*
*oup. But
fl2 ~=Z=3k-1 x Z=3k-1 is normal in B(3, r; 1, 0, 0) hence the Sylow 3-subgroup *
*of Out F(fl2)
has size 9, but that contradicts OutF (fl2) GL 2(3).
It remains to study the case B(3, r; 0, fl, 0). In this case we obtain satura*
*ted fusion systems
with proper F-Alperin subgroups.
p-LOCAL FINITE GROUPS OF RANK TWO FOR ODD p *
* 19
Notation. In what follows we consider the following notation:
o Fix the following elements in Aut(B(3, r; fi, fl, 0)):
- j an element of order two which fixes E0 and permutes E1 with E-1 an*
*d such
that projects to 100-1in Out(B(3, r; fi, fl, 0)).
- ! an element of order two which commutes with j, which projects to -*
* Idin
Out (B(3, r; fi, fl, 0)), and such that fixes Ei for i 2 {-1, 0, 1}.
o By N .flW we denote an extension of type N . W such that its Sylow 3-sub*
*group is
isomorphic to B(3, r; 0, fl, 0).
With all that information now we get the tables with the possible fusion syst*
*ems over
B(3, r; 0, fl, 0) which are not the normalizer of the Sylow 3-subgroup:
Theorem 5.9. Let B be a rank two 3-group of maximal nilpotency class of order a*
*t least 34,
and let (B, F) be a saturated fusion system with at least one proper F-Alperin *
*subgroup. Then
it must correspond to one of the cases listed in the following tables.
o If B ~=B(3, 4; 0, 0, 0) then the outer automorphism group of the F-Alper*
*in subgroups
are in the following table:
_________________________________________________________________
| B | |E0 |E1 | E-1 | V0 | V1 | V-1 | | p-lfg |
|_______|_|_____|____|______|_______|_______|______|_|____________|
| | | | | |SL 2(3) | - | - | |F(34, 1) |
| | | | | |________|_______|______|_|__________ |
| | |- | - | - | - |SL2(3) |SL2(3) | |F(34, 2) |
| | | | | _|_______|_______|_______|_|_________ |
| | | | | |SL 2(3) |SL2(3) |SL2(3) | |L (q1) |
|_______|_|____|______|_____|________|_______|_______|_|_3_______ |
| S|L|2(3) | - | - | - | |E0 .0 SL2(3) |
|________|_|_____|__________|________|_____________|_|___________ |
| | | | | - | SL2(3) | |F(34, 2).2 |
| | | | |________|______________|_|__________ |
| | |- | - |GL 2(3) | - | |F(34, 1).2 |
| | | | | _|_____________|_|___________ |
| | | | | | SL2(3) | |L (q1) : 2 |
| __|_|___|____________|_______|_______________|_|_3________ |
| |G|L2(3) | - | - | - | |E0 .0 GL2(3) |
| | | | | _|_____________|_|___________ |
| | | | | | SL2(3) | |3D4(q2) |
|_______|_|____|____________|_______|_______________|_|__________ |
Table 2. s.f.s. over B(3, 4; 0, 0, 0).
Where q1 and q2 are prime powers such that 3(q1 1) = 2 and 3(q22- *
*1) = 1.
o If B ~=B(3, 4; 0, 2, 0) then the outer automorphism group of the F-Alper*
*in subgroups
are in the following table:
20 ANTONIO D'IAZ, ALBERT RUIZ, AND ANTONIO VIRUEL
_____________________________________
| B | |E0 | V0 | | p-lfg |
|_______|_|_____|______|_|____________|
| | |- |SL2(3) | |F(34, 3) |
|_______|_|_____|_______|_|__________ |
| S|L|2(3) | - |E|0.2 SL2(3) |
|________|_|_____|______|_|__________ |
| | |- G|L2(3) | |F(34, 3).2 |
| __|_|_____|______|_|__________ |
| |G|L2(3) |- | |E0 .2 GL2(3) |
|_______|_|______|_____|_|___________ |
Table 3. s.f.s. over B(3, 4; 0, 2, 0).
o If B ~=B(3, 2k; 0, 0, 0) with k 3 then the outer automorphism group of*
* the F-Alperin
subgroups are in the following table:
_________________________________________________________________
| B | |E0 |E1 |E-1 | V0 | V1 |V-1 | | p-lfg |
|_______|_|_____|___|____|________|_______|_____|_|_______________|
| | | | | |SL2(3) | - | - | |F(32k, 1) |
| | | | | |_______|_______|_______|_|____________ |
| | |- | - | - | - |SL2(3) |SL2(3) | |F(32k, 2) |
| | | | | |_______|_______|_______|_|____________ |
| | | | | |SL2(3) |SL2(3) |SL2(3) | | L (q1) |
|_______|_|____|____|_____|_______|_______|_______|_|__3_________|
| S|L|2(3) | - | - | - | |3 .0 PGL 3(q2) |
|________|_|_____|________|_______|______________|_|_____________ |
| | | | | - | SL 2(3) | |F(32k, 2).2 |
| | | | |_______|_______________|_|____________ |
| | |- | - |GL2(3) | - | |F(32k, 1).2 |
| | | | | |______________|_|_____________ |
| | | | | | SL 2(3) | |L (q1) : 2 |
| __|_|___|__________|_______|_______________|_|_3__________ |
| |G|L2(3) | - | - | - |3|.0PGL 3(q2).2 |
| | | | | |______________|_|_____________ |
| | | | | | SL 2(3) | | 3D4(q3) |
|_______|_|____|__________|_______|_______________|_|____________ |
Table 4. s.f.s. over B(3, 2k; 0, 0, 0) with k 3.
Where qi are prime powers such that 3(q1 1) = k, 3(q2 - 1) = k - 1*
* and
3(q23- 1) = k - 1.
o If B ~=B(3, 2k; 0, fl, 0) with k 3 and fl = 1, 2, then the outer autom*
*orphism group
of the F-Alperin subgroups are in the following table:
_______________________________________
| B | |E0 | V0 | | p-lfg |
|_______|_|_____|______|_|______________|
| | |- |SL2(3) |F|(32k, 2 + fl) |
|_______|_|_____|_______|_|____________ |
| S|L|2(3) | - | |3 .flPGL 3(q) |
|________|_|_____|______|_|____________ |
| | |- G|L2(3) |F|(32k, 2 + fl).2 |
| __|_|_____|______|_|____________ |
| |G|L2(3) |- | |3 .flPGL 3(q) . 2 |
|_______|_|______|_____|_|_____________ |
Table 5. s.f.s. over B(3, 2k; 0, fl, 0) with fl = 1, 2 and k 3.
Where q is a prime power such that 3(q - 1) = k - 1.
p-LOCAL FINITE GROUPS OF RANK TWO FOR ODD p *
* 21
o If B ~= B(3, 2k + 1; 0, 0, 0) with k 2 then the outer automorphism gro*
*up of the
F-Alperin subgroups are in the following table:
____________________________________________________________________
| B | |V0 |V1 | V-1 | E0 | E1 | E-1 | fl1 | | p-lfg |
|_____|_|_____|___|_____|______|_______|______|________|_|___________|
| | | | | |SL2(3) | - | - | - |3|.F(32k, 1) |
| | | | | |_______|______|_______|_______|_|__________ |
| | |- |- |- | - |SL2(3)S|L2(3) | - |3|.F(32k, 2) |
| | | | | |_______|_______|______|_______|_|__________ |
| | | | | |SL2(3) |SL2(3)S|L2(3) | - | |3 . L (q1) |
|______|_|____|____|____|_______|_______|______|_______|_|_____3____ |
| | |- | - | - | - |SL 2(3) |f|l1 : SL2(3) |
|_______|_|_____|_______|_______|_____________|________|_|__________ |
| |S|L2(3) | - | - | - | - | |PGL3(q2) |
|______|_|______|_______|_______|_____________|________|_|__________|
| | | | | | - |GL 2(3) |f|l1 : GL 2(3) *
* |
| | | | | |______________|________|_|__________ *
* |
| | | | | - | SL2(3) | - |3|.F(32k, 2).2 |
| | | | | | _|_______|_|__________ |
| | | | | | |GL 2(3) |F|(32k+1, 1) |
| | | | |______|______________|________|_|__________ |
| | |- | - | | - | - |3|.F(32k, 1).2 |
| | | | | | |________|_|__________ |
| | | | |GL2(3) | |GL 2(3) | |2F4(q3) |
| | | | | _|_____________|________|_|__________|
| | | | | | SL2(3) | - |3|.L (q1) : 2 |
| | | | | | _|_______|_|__3_______ |
| | | | | | |GL 2(3) |F|(32k+1, 2) |
| |_|____|_________|______|______________|________|_|__________ |
| | | | | | - | - |P|GL3(q2) .2 |
| | | | | | |________|_|__________ |
| |G|L2(3) | - | - | |GL 2(3) |F|(32k+1, 3) |
| | | | | _|_____________|________|_|__________ |
| | | | | | SL2(3) | - |3|. 3D4(q4) |
| | | | | | _|_______|_|__________ |
| | | | | | |GL 2(3) |F|(32k+1, 4) |
|______|_|____|_________|______|______________|________|_|__________ |
Table 6. s.f.s. over B(3, 2k + 1; 0, 0, 0) with k 2.
Where qi are prime powers such that 3(q1 1) = k, 3(q2- 1) = k, 3(q*
*23- 1) = k
and 3(q24- 1) = k - 1.
o If B ~=B(3, 2k + 1; 0, 1, 0) with k 2 then F is the saturated fusion s*
*ystem associated
to the group fl1 : SL2(3).
The last column of all this tables gives either the information about the group*
*s which have the
corresponding fusion system, either a name encoded as F(3r, i) to refer to an e*
*xotic 3-local
finite group or the expression of that 3-local finite group as extension of ano*
*ther 3-local finite
group.
Remark 5.10. As particular cases of the classification we find the exotic fusio*
*n systems over
3-groups of order 34 which were announced previously by Broto-Levi-Oliver in [1*
*2, Section 5].
Proof. We divide the proof in three parts:
Classification: In this part we describe the different possibilities for the sa*
*turated fusion
system (B, F) by means of the F-Alperin subgroups and their outer automorphisms*
* groups
Out F(P ). Because AutP (P ) AutF (P ) by Definition 2.1, Out F(P ) = AutF (P*
* )= AutP(P )
also determines AutF (P ), and by Theorem 2.5, these subgroups of automorphisms*
* describe
completely the category F.
22 ANTONIO D'IAZ, ALBERT RUIZ, AND ANTONIO VIRUEL
By hypothesis we have a proper F-Alperin subgroup in B(3, r; fi, fl, ffi), so*
* by Lemma 5.1
ffi = 0, and using now Lemma 5.8 also fi = 0. So we just have to cope with B(3,*
* r; 0, fl, 0).
First of all obverve that fixed i 2 {-1, 0, 1}, Eiand Vicannot be at the same*
* time F-Alperin
subgroups: if Ei is F-Alperin then Vi is F-conjugate to **, so Vi is not F*
*-centric.
Notice that a saturated fusion systems with OutF (B(3, r; 0, fl, 0)) = 1 cann*
*ot contain any
proper F-Alperin subgroup: if it had a proper F-Alperin subgroup, by Lemma 5.7 *
*we would
have a nontrivial morphism in OutF (B(3, r; 0, fl, 0)).
We now begin the analysis depending on the parity of r.
Case r = 2k. Suppose that fl = 0. If OutF (B) = then it is immediate from*
* Lemmas 5.6
and 5.7 that Vi may be F-Alperin but not Ei. The reason is that -100-1 2 SL2(3*
*) lifts to
! 2 OutF (B) = from OutF (Vi) (it is admissible and the lifting does exist)*
*, while it would
lift to j! =2Out F(B) from OutF (E0) (it is admissible but the lifting does not*
* exist), and it
does not lift for Ei with i = -1, 1 (it is not admissible). It is also deduced *
*from Lemmas 5.6
and 5.7 that no Vi can be F-conjugate to Vj for i 6= j because inner conjugatio*
*n in B does
not move {V-1, V0, V1} and outer conjugation is induced just by j and j!. If Vi*
*is F-Alperin
then Out F(Vi) must equal SL2(3) because otherwise we would have a nontrivial e*
*lement in
Out F(B) different from ! for V0, or we would have a non admissible map in E-1 *
*or E1 for V-1
or V1 respectively. So one, two or three of the Vi's can be F-Alperin. A symmet*
*ry argument,
obtained by conjugation by B (see remarks after Lemma 5.5), yields the first th*
*ree rows of
tables 2 and 4. Now assume OutF (B) = . Looking at Lemmas 5.5, 5.6 and 5.7 *
*we obtain
that only E0 can be F-Alperin and moreover, OutF (E0) = SL2(3) because otherwis*
*e OutF (B)
would be Z=2 x Z=2. This is the fourth row of tables 2 and 4. For OutF (B) = there is
no chance for Ei or Vi to be F-Alperin. It remains to cope with the case OutF (*
*B) = .
First, from the argument above E1 and E-1 cannot be F-Alperin. Second, recall f*
*rom the
beginning of this proof that, for i fixed, Eiand Vicannot be F-Alperin simultan*
*eously. Third,
notice that j (and j!) swaps V1 and V-1, so they are F-conjugate. Lastly, from *
*Lemma 5.5,
the only possibility for the outer automorphism groups of E0 and V0 is GL 2(3) *
*in case they
are F-Alperin. Analogously if Vi is F-Alperin, for i = 1, then Out F(Vi) must *
*be SL2(3).
Now a case by case checking yields the last five entries of tables 2 and 4. If *
*r > 4 and fl = 1, 2
(or r = 4 and fl = 2) recall from the conditions in the table of Lemma 5.2 that*
* only E0 and V0
are allowed to be F-Alperin. Similar arguments to those above lead us to tables*
* 3 and 5.
Case r = 2k + 1. For fl = 0 the fusion systems in the table 6 are obtained b*
*y similar
arguments to those of the preceding cases, bearing fl1 may be F-Alperin too in *
*mind. To fill
in this table, notice that if some Eior Viis F-Alperin then - Idis an outer aut*
*omorphism of
this group that must lift, following Lemma 5.6, to ! or j! for Ei and Vi respec*
*tively. But !
and j! restrict in OutF (fl1) to automorphisms of determinant -1, so in case fl*
*1 is F-Alperin,
when some Eior Viis so, its outer automorphism group must be OutF (fl1) = GL 2(*
*3). Notice
that when fl1 is F-Alperin, Out F(B) must contain j by Lemma 5.6 and so E-1 and*
* E1
are F-conjugate. In case of fl = 1, Lemmas 5.5 and A.14 imply that only 100-1 *
*can be in
Out F(B). If fl1, E0 or V0 were F-Alperin then OutF (B) would contain j, ! or j*
*! respectively.
So the only chance is that fl1 is the only F-Alperin subgroup. Notice also that*
* OutF (fl1) must
equal SL2(3), because in other case there would be a non-trivial element in Out*
*F (B) different
from j.
p-LOCAL FINITE GROUPS OF RANK TWO FOR ODD p *
* 23
Saturation: Now we prove that all the fusion systems obtained in the classifica*
*tion part of
this proof are saturated by means of [12, Proposition 5.3]. To show that a cert*
*ain fusion system
(B, F) appearing in the tables is saturated the method consists in setting G de*
*f=B : OutF (B),
where OutF (B) is the entry for B in the table, and for each G-conjugacy class *
*of F-Alperin
subgroups choosing a representative P B and setting KP def=OutG(P ) (KP is de*
*termined
by Lemma 5.6) and P def=OutF(P ), where OutF (P ) is the entry for P in the ta*
*ble. In order
to obtain that the fusion system under consideration is saturated, for each cho*
*sen F-Alperin
subgroup P B it must be checked that:
(1)P does not contain any proper F-centric subgroup.
(2)p - [ P : KP] and for each ff 2 P \ KP, KP \ ff-1KPff has order prime t*
*o p.
On the one hand, by Lemma 5.2 the first condition is fulfilled by all the fusio*
*n systems
obtained in the classification part of this proof. On the other hand, it is ver*
*ified that OutB (P )
equals < 1011>, < 1101> and < 1101> for fl1, Eiand Virespectively. Denoting *
*by ~P this order 3
outer automorphism it is an easy check that for all the fusion systems in the t*
*ables:
o P is F-Alperin with P = SL2(3) just in case KP = <~P, -100-1>,
o P is F-Alperin with P = GL 2(3) just in case KP = <~P, -1001, 100-1>.
Now the two pairs (KP, P) above verify the condition (2).
Notice that the group G = B : OutF (B) defined earlier can be constructed bec*
*ause 3 does
not divide the order of OutF (B) (see Remark 2.3) and the projection Aut(B) i O*
*ut(B) has
kernel a 3-group, and thus there is a lifting of Out F(B) to Aut(B). A more del*
*icate point
in the proofs of classification and saturation is that (recall the remarks afte*
*r the Lemma 5.3)
the outer automorphisms groups Out F(P ) for P B F-Alperin (P can be the whol*
*e B)
appearing in the tables are described as subgroups of GL 2(3), and that while f*
*or P = Ei, Vi
j
the Frattini maps Out(P ) ! GL 2(3) are isomorphisms, for fl1 and B they are no*
*t.
The choice of SL2(3) and GL 2(3) lying in Out(fl1) = Aut(fl1) and of Z=2 and *
*Z=2 x Z=2
Out (B) are not totally arbitrary. In fact, the choice of AutF (B) must go by t*
*he restriction
map Aut(B) ! Aut(fl1) to the choice of AutF (fl1). Moreover, as fl1 is characte*
*ristic in B and
diagonal automorphisms in Aut F(fl1) = Out F(fl1) must lift to Aut F(B) (Lemma *
*5.7), one
can check that the choice of AutF (fl1) determines completely the choice of Out*
*F (B).
Now we prove that different choices gives isomorphic saturated fusions system*
*s. Let (B, F)
and (B, F0) be saturated fusion systems that correspond to the same row in some*
* table
of the classification. Suppose first that fl1 is not F-Alperin: the two semid*
*irect products
H = B : OutF (B) and H0 = B : OutF0(B) are isomorphic because the lifts of OutF*
* (B) and
Out F0(B) to Aut(B) are Aut(B)-conjugate as the projection Aut(B) i GL 2(3) has*
* kernel
a 3-group. Then we have an isomorphism of categories FB (H) ~= FB (H0) which c*
*an be
extended to an isomorphism F ~=F0.
If fl1 is F-Alperin then (recall that AutF (fl1) determines OutF (B)) we buil*
*d the semidirect
products H = fl1 : Out F(fl1) and H0 = fl1 : Out F0(fl1). These groups are iso*
*morphic by
Corollary A.19 and have Sylow 3-subgroup B. Then we have an isomorphism of cate*
*gories
FB (H) ~=FB (H0) which can be extended to an isomorphism F ~=F0.
24 ANTONIO D'IAZ, ALBERT RUIZ, AND ANTONIO VIRUEL
Exoticism: To justify the values in the last column we have to cope with the po*
*ssible finite
groups with the fusion systems described there.
Consider first all the fusion systems in the tables such that they have at le*
*ast one F-
Alperin rank two elementary abelian subgroup (respectively at least one F-Alper*
*in subgroup
isomorphic to 31+2+and also fl1 is F-Alperin). Consider N fi B(3, r; 0, fl, 0*
*) a nontrivial
proper normal subgroup which is strongly closed in F. By Lemma A.10, N must co*
*ntain
the center of B, and as there is an F-Alperin rank two elementary subgroup N mu*
*st also
contain s (respectively, if fl1 is F-radical N must also contain flr-2, and as *
*there is an F-
Alperin subgroup isomorphic to 31+2+, N must contain s too). Again by Lemma A.1*
*0 N must
be isomorphic to 31+2+if r = 4 or B(3, r - 1; 0, 0, 0) if r > 4.
In all these cases we can apply Proposition 2.19, getting that if they are th*
*e fusion system
of a group G, then G can be choosen to be almost simple. Moreover the 3-rank of*
* G and the
simple group of which G is an extension must be two, so we have to look at the *
*list of all the
simple groups of 3-rank two:
a) The information about the sporadic simple groups can be deduced from [25, Ta*
*bles 5.3 &
5.6.1], getting that all of the groups in that family which 3-rank equals tw*
*o have Sylow
3-subgroup of order at most 33, and there are not outer automorphisms of ord*
*er 3.
b) The p-rank over the Lie type simple groups in a field of characteristic p ar*
*e in [25, Ta-
ble 3.3.1], where taking p = 3 and the possibilities of the groups of 3-rank*
* two one gets
that the order of the Sylow 3-subgroup is at most 33, and again there are no*
*t outer auto-
morphism of order 3.
c) The Lie type simple groups in characteristic prime to 3 have a unique elemen*
*tary 3-
subgroup of maximal rank, out of L3(q) with 3|q - 1, L-3(q) with 3|q + 1, G2*
*(q), 3D4(q)
or 2F4(q) by [23, 10-2]. So, as 31+2+and B(3, r; 0, fl, 0) do not have a un*
*ique elementary
abelian 3-subgroup of maximal rank, we have to look at the fusion systems of*
* this small
list.
The fusion systems induced by L+3(q), when 3|(q - 1), and by L-3(q), when *
*3|(q + 1), are
the same, and can be deduced from [13, Example 3.6] and [5] using that there*
* is a bijection
between radical subgroups in SL3(q) and radical subgroups in PSL 3(q), so ob*
*taining the
result in Table 4. The extensions of L3(q) by a group of order prime to 3 mu*
*st be also
considered, getting fusion systems over the same 3-group. Finally, Out (L3(*
*q)) has 3-
torsion, so we must consider the possible extensions, getting the group PGL *
*3(q) and an
extension PGL 3(q).2. The study of the proper radical subgroups in this case*
* is done in [5],
getting that the only proper F-radical is V0.
The fusion system of G2(q) is studied in [28] and [16], getting that it co*
*rresponds to the
fusion system labeled as 3L+3(q) : 2.
The fusion system of 3D4(q) can be deduced from [27], getting the desired *
*result.
Finally the fusion system of 2F4(q) has been studied in [13, Example 9.7].
This classification tells us that all the other cases where there is an F-Alp*
*erin rank two
elementary abelian 3-subgroup, and also the ones such that fl1 and one subgroup*
* isomorphic
to 31+2+are F-radical, must correspond to exotic p-local finite groups.
p-LOCAL FINITE GROUPS OF RANK TWO FOR ODD p *
* 25
Consider now the cases where the only proper F-Alperin subgroup is fl1. In th*
*ose cases it
is straightforward to check that they correspond to the groups fl1 : SL2(3) and*
* fl1 : GL 2(3),
where the actions are described in Lemma A.17.
In all of the remaining cases, the ones where all the proper F-Alperin subgro*
*ups are
isomorphic to 31+2+, the normalizer of the center of B(3, r; 0, fl, 0) in F is *
*the whole fusion
system F (i.e. Z(B(3, r; 0, fl, 0)) is normal in F).
Consider first the ones where Z(B(3, r; 0, fl, 0)) is central in F, i.e. the*
* ones where for
all Ei proper F-radical subgroup isomorphic to 31+2+we have Out F(Ei) = SL 2(3)*
*. Using
Lemma 2.21, we get that they correspond to groups if and only if the quotient b*
*y the center
corresponds also to a group, getting again the results in the tables.
Finally it remains to justify that 3F(32k, 1).2 and 3F(32k, 2).2 are not the *
*fusion system
of finite groups. Consider G a finite group with one of those fusion systems, *
*and consider
Z(B) the center of B(3, 2k + 1; 0, 0, 0). Consider now the fusion system constr*
*ucted as the
centralizer of Z(B) (Definition 2.11) then, by Remark 2.12 3F(32k, 1) or 3F(32k*
*, 2) would be
also the fusion system of the group CZ(B)(G), and we know that these are exotic.
Appendix A. Rank two p-groups
In this appendix we recall all the information and properties of p-rank two p*
*-groups that
we need to classify the saturated fusion systems over these groups.
The classification of the rank two p-groups, p > 2, traces back to Blackburn *
*(e.g. see [29,
Theorem 3.1]):
Theorem A.1. Let p be an odd prime. Then the p-groups of p-rank two are the one*
*s listed
here:
(i)The non-cyclic metacyclic p-groups, which we denote M(p, r).
(ii)The groups C(p, r), r 3 defined by the following presentation:
def p p pr-2 pr-3
C(p, r) = .
(iii)The groups G(p, r; ffl), where r 4 and ffl is either 1 or a quadratic n*
*on-residue modulo p
defined by the following presentation:
def p p pr-2 -1 fflpr-3
G(p, r; ffl) = .
(iv)If p = 3 the 3-groups of maximal nilpotency class, except the cyclic group*
*s and the wreath
product of Z=3 by itself.
Where [x, y] = x-1y-1xy.
Proof. According to [20, Theorem 5.4.15], the class of rank two p-groups (p odd*
*) agrees
with the class of p-groups in which every maximal normal abelian subgroup has r*
*ank two,
or equivalently every maximal normal elementary abelian subgroup has rank two. *
* As the
only group of order p3 which requires at least three generators is (Z=p)3, the *
*class of rank
two p-groups (p odd) agrees with the class of p-groups in which every normal su*
*bgroup of
size p3 is generated by at most two elements. The latter class of p-groups is d*
*escribed in [7,
Theorem 4.1] when the order of the group is pn for n 5 (and p > 2) while the *
*case n 4
can be deduced for the classification of p-groups of size at most p4 [14, p. 14*
*5-146].
26 ANTONIO D'IAZ, ALBERT RUIZ, AND ANTONIO VIRUEL
To complete the classification above we also need a description of maximal ni*
*lpotency class
3-groups, which is given in [6, last paragraph p. 88]:
Theorem A.2. The non cyclic 3-groups of maximal nilpotency class and order grea*
*ter than 33
are the groups B(3, r; fi, fl, ffi) with (fi, fl, ffi) taking the values:
o For any r 5, (fi, fl, ffi) = (1, 0, ffi), with ffi 2 {0, 1, 2}.
o For even r 4, (fi, fl, ffi) 2 {(0, fl, 0), (0, 0, ffi)}, with fl 2 {1,*
* 2} and ffi 2 {0, 1}.
o For odd r 5, (fi, fl, ffi) 2 {(0, 1, 0), (0, 0, ffi)}, with ffi 2 {0, *
*1}.
With these parameters, B(3, r; fi, fl, ffi) is the group of order 3r defined by*
* the set of generators
{s, s1, s2, . .,.sr-1} and relations
si= [si-1, s] for i 2 {2, 3, . .,.r - 1}, *
* (1)
[s1, s2] = sfir-1, *
* (2)
[s1, si] = 1 for i 2 {3, 4, . .,.r - 1}, *
* (3)
s3 = sffir-1, *
* (4)
s31s32s3 = sflr-1, *
* (5)
s3is3i+1si+2= 1 for i 2 {2, 3, . .,.r - 1}, and assuming sr = sr+1=.*
*1(6)
Remark A.3. For p = 3 and r = 4 we have that B(3, 4; 0, 0, 0) ~=G(3, 4; 1), B(3*
*, 4; 0, 2, 0) ~=
G(3, 4; -1) and B(3, 4; 0, 1, 0) is the wreath product 3 o 3 that has 3-rank th*
*ree.
Remark A.4. In [6] the classification of the p-groups of maximal rank depends o*
*n four
parameters ff, fi, fl and ffi, but for p = 3 we have ff = 0. In all the paper,*
* with the no-
tation B(3, r; fi, fl, ffi) we assume that the parameters (fi, fl, ffi) corresp*
*ond to the stated in
Theorem A.2 for rank two 3-groups.
What follows is a description of the group theoretical properties of the grou*
*ps listed in
Theorems A.1 and A.2, that are used along the paper.
We begin with the family C(p, r):
Lemma A.5. Consider C(p, r) as in Theorem A.1, with the same notation for the g*
*enerators:
(a) The center is ~=Z=pr-2.
r-3
(b) The commutators are determined by [aibj, asbt] = c(it-sj)p.
(c) C(p, r) = Z(C(p, r)) 1(C(p, r)) and therefore C(p, r) is isomorphic to the *
*central product
Z=pr-2O C(p, 3).
(d) The restriction of the elements in Aut(C(p, r)) to Z(C(p, r)) and 1(C(p, r*
*)) provides an
isomorphism
r-3 r-2
Aut(C(p, r)) ~= Z=p x ASL 2(p) : (p - 1) < Aut(Z=p ) x Aut(C(p, 3))
that gives rise to a group epimorphism æ: Out (C(p, r)) ! GL 2(p) which map*
*s a morphism
0j0k0 i i0
of type a 7! aibjck, b 7! aib c to the matrix j j0.
(e) Aut(C(p, r)) = Inn(C(p, r)) o Out(C(p, r)) ~=(Z=p x Z=p) o (Z=pr-3x GL 2(p)*
*).
(f) For G = GL 2(p) or SL2(p), the group Out(C(p, r)) contains just one subgrou*
*p isomorphic
to G up to conjugation, but in the case G = SL2(3) and r > 3. The group Out*
* (C(3, r)),
r > 3, contains two different conjugacy classes of subgroups isomorphic to *
*SL2(3).
p-LOCAL FINITE GROUPS OF RANK TWO FOR ODD p *
* 27
Proof. The statement (a) follows from the presentation of C(p, r) while (b) can*
* be read from
[19, Lemma 1.1]. The central product given in (c) is obtained by identifying Z=*
*pr-2with =
r*
*-3
Z(C(p, r)) and C(p, 3) with = 1(C(p, r)), so their intersection is = Z().
Moreover, Z(C(p, r)) and 1(C(p, r)) are characteristic subgroups of C(p, r), a*
*nd therefore
every element in Aut(C(p, r)) maps each of these subgroups to itself. Then (c) *
*implies that
r-3 pr-3
Aut(C(p, r)) = {(f, g) 2 Aut() x Aut()| f(cp ) = g(c )}
providing the description of Aut(C(p, r)) in (d) while the morphism æ is obtain*
*ed by consid-
ering outer automorphisms and projecting on the C(p, 3) factor (see [35, Lemma *
*3.1]). As
Aut (G) = Inn(G) o Out(G) for G = Z=pr-2, C(p, 3), (d) implies (e). Finally, no*
*tice that if
G = GL 2(p) or SL2(p), GL 2(p) contains just one copy of G up to conjugacy, and*
* therefore, the
number of subgroups of Out(C(p, r)) ~=Z=pr-3x GL 2(p) isomorphic to G up to con*
*jugation
depends on Hom (G, Z=pr-3). Since G is p-perfect for p > 3, the latter set cont*
*ains just one
element unless p = 3, r > 3 and G = SL2(3). Finally, Hom (SL 2(3), Z=3r-3) cont*
*ains three
elements that give rise to three subgroups of type SL 2(3) in Out (C(3, r)), de*
*termined by
their Sylow 3-subgroups Sidef=<(3r-4i, 1101)> for i = 0, 1, 2 (notice that SL2*
*(p) is generated
by elements of order p [20, Theorem 2.8.4]). But S1 and S2 are conjugate in Ou*
*t (C(3, r))
and therefore there are just two conjugacy classes of SL2(3) in Out(C(3, r)) if*
* r > 3.
Lemma A.6. Let H be a p-centric subgroup of C(p, r), then H is either the total*
* or H ~=
Z=p x Z=pr-2.
Proof. Assume H is a p-centric centric subgroup of C(p, r). So H must contain t*
*he center
as a proper subgroup. Let aibj 2 H \ , then we have that ~=Z=p x Z*
*=pr-2 is a
self-centralizing maximal subgroup in C(p, r), so then H is either the total or*
* .
The following properties of G(p, r; ffl) can be deduced directly from [19, Se*
*ction 1]
Lemma A.7. Consider G(p, r; ffl) as in Theorem A.1, with the same notation for *
*the gener-
ators:
(a) The commutators are determined by the formula [aibjck, asbtcu] = biu-skcn w*
*here n =
fflpr-3(ui(i-1)_2+ js - it - k s(s-1)_2).
*
* r-3
(b) The center of G(p, r; ffl) is the group generated by , and, as r 4, i*
*t contains .
r-3*
* t u
(c) There is a unique automorphism in G(p, r; ffl) which maps æ(a) = aibjclp a*
*nd æ(c) = b c
for any i, j, l, t, u 2 { 1} x Z=p x Z=p x Z=p x (Z=pr-2)*.
28 ANTONIO D'IAZ, ALBERT RUIZ, AND ANTONIO VIRUEL
Lemma A.8. If (p, r; ffl) 6= (3, 4; 1), the p-centric subgroups of G(p, r; ffl)*
* are the ones in the
following table:
_____________________________________________________
| Isomorphism type | Subgroup |
|__________________|_________________________________|_
| G(p, r; ffl) | |
| | |
| Z=p x Z=pr-2 | *** |
| | |
| Z=pr-2 |with i 2 Z=p and j 2 (Z=pr-2)*|
| | |
| M(p, r - 1) | with j 2 (Z=pr-2)* |
| | |
| Z=p x Z=pr-3 | |
| | |
| C(p, r - 1) | |
|___________________|________________________________ |
Proof.It is clear that the total is a p-centric subgroup, so let H < G(p, r; ff*
*l) be a p-centric
subgroup different from the total. As it must contain its centralizer in G(p, r*
*; ffl) we have that
< H.
We divide the proof in different cases:
Case H ****: then, as **** is commutative, we have H = **** ~=Z=p x Z=p*
*r-2, and
using the commutator rules of this group one can check that it is p-centric.
So in the following cases there is an element of the form ff = abicj.
Case p - j and H cyclic: as p - j we can construct an automorphism of G(p, r; f*
*fl) sending
a 7! abi and c 7! cj, so we can compute the order of ff computing the order of *
*ac. Now one
can check the following formula:
n n- n fflpr-3
(ac)n = anb-(2)c (3) .
So if (p; r, ffl) 6= (3; 4, 1) we get that (ac)p = c p, so ac has order pr-2 an*
*d it is self-centralizing,
so it is p-centric. In this case we have H = ~=Z=pr-2.
Case p - j and H not cyclic: we can assume that H has two generators, and one o*
*f them
is ff: if we consider an element fi of H \ then the order of is a*
*t least pr-1, so if
we add another element we would have the total. As we are considering that H is*
* not the
total, we can assume H = . We can also assume that fi = bkcl(if the gen*
*erator a would
appear in the expression of fi we could take a power of fi and multiply it by f*
*f-1 to change
the generators). So consider H = with H not cyclic and different from *
*the total, then
we prove that it is metacyclic: the order of H is pr-1 and, as H is not cyclic,*
* then fi 62 ,
so either k 6= 0 or p - l. If k = 0 then, as p - l we have H = , and as*
* [abi, c] = b, H
is the total, which is not considered. So k 6= 0 and now we have to distinguish*
* between two
cases: p|l and p - l. If p - l we can consider the inverse of the automorphism *
*æ in G(p, r; ffl)
with æ(a) = a and æ(c) = bkcland we have æ-1(H) = = G(p, r; ffl), so*
* H = G(p, r; ffl),
that implies that p|l and we can consider the group generated by .*
* One more
reduction is cancellation of bi by means of a multiplication by (bcpf)-i, so th*
*e generators are
ff = acj and fi = bcpf. We can still simplify the generators using that =*
* , getting
r-3
. To see that it is metacyclic just check that [H, H] = ~= Z=p,*
* and that the
class of ff in the quotient H=[H, H] has order pr-3, the same order as H=[H, H]*
*, so it is cyclic,
and H ~=M(p, r - 1).
p-LOCAL FINITE GROUPS OF RANK TWO FOR ODD p *
* 29
Case p | j: we can assume that all the elements in H are of the form akblcpm, b*
*ecause if there
were an element in H with an exponent in c which is not multiple of p then we w*
*ould be in
one of the previously studied cases. As cp must be in H, then we have that H.
One can check by means of the commutator formula that the group ~=Z=p*
* x Z=pr-3
is self-centralizing, so p-centric. Moreover if H \ 6= ;, all the oth*
*er restrictions of this
case imply that there is an element of the form ajbk in H \ , and an e*
*asy calculation
gives us that = ~=C(p, r - 1). This also proves that*
* there is only one
p-centric subgroup in G(p, r; ffl) isomorphic to C(p, r - 1).
It remains to study maximal nilpotency class 3-groups, beginning with the fol*
*lowing prop-
erties that can be read in [6] and [26, III.x14]:
Proposition A.9. Consider B(3, r; fi, fl, ffi) as defined in Theorem A.2 with t*
*he same notation
for the generators. Then the following hold:
(a) From relations (1) to (6) we get:
2
(s 1s``11.s.`.`r-1r-1)3 = srffi+fl``1-fi``11. *
* (7)
def *
* r-i
(b) fli(B(3, r; fi, fl, ffi)) = are characteristic subgr*
*oups of order 3 gener-
ated by si and si+1for i = 1, .., r - 1 (assuming sr = 1).
(c) fl1(B(3, r; fi, fl, ffi)) is a metacyclic subgroup.
(d) fl1(B(3, r; fi, fl, ffi)) is abelian if and only if fi = 0 .
(e) The extension
1 ! fl1 ! B(3, r; fi, fl, ffi) i!Z=3 ! 1
is split if and only if ffi = 0.
(f) Z(B(3, r; fi, fl, ffi)) = flr-1(B(3, r; fi, fl, ffi)) = .
The following lemma is useful when studying the exoticism of the fusion syste*
*ms con-
structed in Section 5.
Lemma A.10. Let N be a nontrivial proper normal subgroup in B(3, r; 0, fl, 0). *
*Then
(a) N contains Z(B(3, r; 0, fl, 0)).
(b) If N contains s then N ~=31+2+if r = 4 or N ~=B(3, r - 1; 0, 0, 0) if r > 4.
Proof. According to [4, Theorem 8.1] N must intersect Z(B(3, r; 0, fl, 0)) in a*
* nontrivial sub-
group. As in our case the order of the center is p, we obtain (a).
From [6, Lemma 2.2] we deduce that if the index of N in B(3, r; 0, fl, 0) is *
*3lwith l 2 then
N = fll, and it does not contain s. So a proper normal subgroup N which contain*
*s s must
be of index 3 in B(3, r; 0, fl, 0), so B(3, r; 0, fl, 0)=N is abelian, and the *
*quotient morphism
B(3, r; 0, fl, 0) ! B(3, r; 0, fl, 0)=N factors through B(3, r; 0, fl, 0)=fl2(B*
*(3, r; 0, fl, 0)) ~=Z=3 x
Z=3 (the commutator of B(3, r; 0, fl, 0) is fl2(B(3, r; 0, fl, 0))), generated *
*by the classes _sand __s1.
Now we have to take the inverse image of the proper subgroups in Z=3xZ=3 of ord*
*er 3, getting
that N must be fl1, **~~, or . Only the second contain*
*s s, getting the
second part of the result.
30 ANTONIO D'IAZ, ALBERT RUIZ, AND ANTONIO VIRUEL
In what follows we restrict to r 5 and the parameters (fi, fl, ffi) 2 {(0, *
*fl, 0), (1, 0, 0)}, with
fl 2 {0, 1, 2}. This includes all the possibilities for the parameters describe*
*d in Theorem A.2
except the cases ffi = 1 and {(0, 0, 0), (0, 2, 0)} for r = 4. We fix k the int*
*eger such that r = 2k
if r is even and r = 2k + 1 if r is odd.
Lemma A.11. For the groups B(3, r; fi, fl, 0) it holds that:
(a) fl2(B(3, r; fi, fl, 0)) is abelian.
(b) The orders of s1, s2 and s3 are 3k, 3k and 3k-1 if r = 2k + 1 and 3k, 3k-1 *
*and 3k-1 if
r = 2k.
Proof.We check that fl2(B(3, r; fi, fl, 0)) is abelian. If fi = 0 then fl1 is a*
*belian and fl2 < fl1,
and if fi = 1 (which implies fl = 0) we have to see that s2 and s3 commute. We*
* have the
following equalities:
(5)3 3 -1 -3 -3 3 -1 -3
[s3, s2] = s-13s-12s3s2 = s1s2s2 s2 s1 s2 = s1s2 s1 s2.
So we have reduced to check that s2 and s31commute.
We use equation (2) to deduce s-12s1s2 = s1sr-1, and raise it to the cubic po*
*wer to get
s-12s31s2 = s31s3r-1. Equation (6) for i = r - 1 tells us that the order of sr-*
*1 is 3, so s2 and s31
commute.
We now compute the orders of s1, s2 and s3. Begin using that sr-1has order 3.*
* Equation (6)
for i = r - 2 yields s3r-2s3r-1= 1, from which sr-2 has order 3 too. For i = r *
*- 3 the equation
becomes s3r-3s3r-2sr-1 = 1, and so sr-3 has order 9. An induction procedure, ta*
*king care of
the parity of r, provides us with the desired result.
The next lemma describes the conjugation by s action on fl1:
Lemma A.12. For B(3, r; fi, fl, 0) the conjugation by s on the characteristic *
*subgroup
fl1(B(3, r; fi, fl, 0)) is given by:
_________________________________________________________________________
| |fi = 0, r = 2k +|1 fi = 0, r = 2k | fi = 1 |
|_______|________________|_____________________|_________________________|_
| | s | s | f s f f f(1-f)=2|
| | s1=s1s2 | s1=s1s2 | (s1 ) =s1 s2 sr-1 |
| fl = 0 | | | |
| | ss=s-3s-2 | ss=s-3s-2 | (s2f)s=s1-3fs2-2f |
|________|__2__1__2______|______2__1__2________|_________________________|
| | s | s | |
| | s1=s1s2 | s1=s1s2 | |
| fl = 1 | (-3)k-1-2| -3((-3)k-2+1) | - |
| |ss=s-3s | ss=s s-2 | |
|________|_2__1__2_______|__2__1___________2___|________________________|_
| | | s | |
| | | s1=s1s2 | |
| fl = 2 | - | 3((-3)k-2-1) | - |
| | | ss=s s-2 | |
|________|_______________|___2__1__________2___|________________________|_
Proof.We begin first with the case of fi = 0. To find the expression for the c*
*onjugation
by s action on fl1 notice that s1s = s1[s1, s] = s1s2 by equation (1) and that *
*analogously
s2s= s2[s2, s] = s2s3. So we need to express s3 as a product of powers of s1 an*
*d s2. We begin
writing sr-1 as a product of powers of s2 and s3. Bearing this objective in min*
*d we use the
same equation (6) as before, but beginning with i = 2. In this case we obtain s*
*4 = s2-3s3-3.
For i = 3 the relation is
s5 = s3-3s4-3= s3-3(s2-3s3-3)-3 = s29s36.
p-LOCAL FINITE GROUPS OF RANK TWO FOR ODD p *
* 31
If i 4 and we got in an earlier stage
si= s2ais3bi,si+1= s2ai+1s3bi+1
then equation (6) reads as
si+2=si-3si+1-3=(s2ais3bi)-3(s2ai+1s3bi+1)-3=s2-3ai-3ai+1s3-3bi-3bi+*
*1.
So sr-1= s2ar-1s3br-1, where ar-1and br-1are obtained from the recursive sequen*
*ces a4 = -3,
a5 = 9, ai+2= -3ai- 3ai+1and b4 = -3, b5 = 6, bi+2= -3bi- 3bi+1for i 4.
*
* 0(3-fla )b0
Substituting this last result in equation (5) in Theorem A.2 we reach s3 = s1*
*3bs2 r-1,
01+(3-fla )b0
where b0(flbr-1- 1) = 1 modulus the order of s3 and s2s= s13bs2 r-1. Fina*
*lly, some
further calculus using the recursive sequences {ai} and {bi} and taking into ac*
*count separately
the three possible values of fl finishes the proof for fi = 0.
For the case fi = 1 use the relations in the presentation of Theorem A.2 to f*
*ind the
commutator rules in B(3, r; 1, 0, 0).
Remark A.13. In the cases with fi = 0 we have that fl1(B(3, r; 0, fl, 0)) is a *
*rank two abelian
subgroup generated by {s1, s2} so we can identify the conjugations by s with a *
*matrix Mr,fls,
obtaining:
2k+1,1 1 -3
Mr,0s= 1-31-2, Ms = 1(-3)k-1-2,
k-2 2k,2 k-2
M2k,1s= 1-3((-3)1-2+1)and Ms = 13((-3)1-2-1).
Next we determine the automorphism groups of B(3, r; 0, fl, 0) and B(3, r; 1,*
* 0, 0).
Lemma A.14. The automorphism group of B(3, r; fi, fl, 0) consists of the homomo*
*rphisms
that send
0 e00
s 7! ses1es2
0 f00
s1 7! sf1s2
where the parameters verify the following conditions:
___________________________________________________________
| | fi = 0, r odd | fi = 0, r even | fi = 1 |
|_______|_______________|_________________|________________|_
| fl = 0 |e = 1, 3 - f0 |e = 1, 3 - f0 e|= 1, 3 | e0, 3 - f0 |
|________|________________|_________________|______________ |
| fl = 1e|= 1, 3 | e0, 3 -ef0|= 1, 3 | e0, 3 - f0|- |
|________|___________________|____________________|________ |
| fl = 2 | - |e = 1, 3 | e0, 3 - f0| - |
|________|______________|______________________|___________|
Proof. We begin with the case of fi = 0. As fl1 = is characteristic an*
*d B(3, r; 0, fl, 0)
is generated by s and s1, every homomorphism ' 2 B(3, r; 0, fl, 0) is determine*
*d by
0 e00 f0 f00
s 7! ses1es2 , s1 7! s1 s2
for some integers e, e0, e00, f0, f00. In the other way, given such a set of pa*
*rameters the equa-
tions (1)-(6) from Theorem A.2 give us which conditions must verify these param*
*eters in
order to get a homomorphism '. In the study of these equations denote by Ms the*
* matrix of
conjugation by s on fl1 described in Remark A.13.
32 ANTONIO D'IAZ, ALBERT RUIZ, AND ANTONIO VIRUEL
o Equation (1): it is straightforward to check that these conditions are e*
*quivalent to
0 f00 f0i f0i-1 *
* e
'(si) = s1fis2ifor every 2 i n - 1 with f00i= Me f00i-1, where Me =*
* Ms - Id,
f01= f0and f001= f00. In particular we obtain the value of '(s2), and so*
* we get the
restriction of ' to fl1 = .
o Equations (2) and (3): as fi = 0 fl1 is abelian and these equations are *
*satisfied if so
are the conditions for equation (1).
o Equation (4): using equation (7) from Proposition A.9 we find that if fl*
* = 0 there are
no additional conditions and if fl = 1, 2 it must be verified that 3 | e*
*0.
o Equation (5): the condition is:
f0
(3 + 3Me+ M2e- flMe r-2) f00= 0.
For fl = 0 it holds that 3 + 3Me + M2e= 0. So there is not more conditio*
*ns on the
parameters. If fl = 1, 2, using the integer characteristic polynomial of*
* Me to compute
Mer-2 we produce two recursive sequences, similar to the ones in Lemma A*
*.12, which
0
(r - 1)-th term equals (3 + 3Me+ M2e- flMe r-2) ff00. Checking details *
*we obtain just
the condition 3 | f0if r is odd, fl = 1 and e = -1.
o Equation (6): it is easy to check that, as r 5, the imposed condition*
*s for i =
2, ..., r - 1 are equivalent to:
f0
Me(3 + 3Me+ M2e) f00= 0.
Because M3s= Id, in fact it holds that Me(3 + 3Me + M2e) = 0. So there *
*are not
additional conditions on the parameters.
So we have just got the conditions on the parameters so that there exists a hom*
*omorphism '
such that the images on s and s1 are predetermined by these parameters e, e0, e*
*00, f0, f00. It
just remains to look out the conditions on ' so it is an automorphism. A quick *
*check shows
that the conditions are 3 - e and the determinant of '|fl1 is invertible modulu*
*s 3, which is
equivalent to 3 - f0.
For B(3, r; 1, 0, 0) the arguments are similar using the commutator rules for*
* powers of s,
s1 and s2.
In the next two lemmas we find out which copies of Z=3n x Z=3n and C(3, n) ar*
*e in
B(3, r; fi, fl, 0) and how they lie inside this group.
Lemma A.15. Let P be a proper p-centric subgroup of B(3, r; 0, fl, 0) isomorphi*
*c to Z=3n x
Z=3n or C(3, n) for some n. Then P is determined, up to conjugation, by the fol*
*lowing table:
_________________________________________________________________
| Isomorphism |Subgroup (up to| |
| | |Conditions |
| type |conjugation) | |
|_____________|_______________|___________________________________|
| Z=3k x Z=3k |fl1 = |r = 2k + 1. |
|_____________|_______________|__________________________________||||
| 1+2 | def |i = s 3k-1, i0= s 3k-1for r =2k+1,|
| 3+ |Ei= ~~*| 2 k-1 1 k-2 |
| | |i = s 3 , i0= s -3 for r = 2k, |
|_____________|________________| 1 2 |
| | def |i 2 {-1, 0, 1} if fl = 0 and i = 0|if
| Z=3 x Z=3 |V = ** | |
| | i 1 |fl = 1, 2. |
|_____________|_______________|__________________________________|
p-LOCAL FINITE GROUPS OF RANK TWO FOR ODD p *
* 33
Proof. Firstly, suppose that P is contained in fl1. As fl1 is abelian and P mu*
*st contain its
centralizer as it is p-centric, P must equal fl1. Recalling the orders of s1 an*
*d s2 we check that
only the case r odd is allowed.
Suppose now that P is not contained in fl1. Then P fits in the short exact se*
*quence:
1 ! K ! P !i Z=3 ! 1,
with K fl1. If K = Z=3m then, as P ~= Z=3n x Z=3n or P ~= C(3, n), we get th*
*at the
0
only possibility is m = 1 and P ~=Z=3 x Z=3. Suppose then that K = Z=3m x Z=3m *
*. Now,
checking cases again, the only chance for P is to be C(3, n). An easy calculati*
*on shows that
CB(3,r;0,fl,0)(P ) ~= Z=3. As this centralizer must contain the center of P ~=*
* C(3, n), which
from Lemma A.5 is Z(C(3, n)) ~=Z=3n-2, n must equal 3 and P ~=31+2+. Now we det*
*ermine
precisely all the p-centric subgroups isomorphic to 31+2+or Z=3 x Z=3, and thei*
*r orbits under
B(3, r; 0, fl, 0)-conjugation.
o We begin with the p-centric subgroups isomorphic to 31+2+. As they are p*
*-centric
they must contain the center of B(3, r; 0, fl, 0) which is equal to **.*
* So, from the
earlier discussion, they are of the form 31+2+adef=** : , where*
* i0 is as in the
statement of the lemma and a 2 fl1 is, in principle, arbitrary. Now, sa *
*having order 3
is seen to be equivalent to N(a) = 0 where N is the norm operator N = 1 *
*+ s + s2.
On the other hand, from the description of Ms in Remark A.13, if a = s1a*
*1s2a2and
0 a0 1+2 1+2 *
* 0
a0= s1a1s22then 3+a and 3+a0 are B(3, r; 0, fl, 0)-conjugate if and onl*
*y if a1 and a1
are congruent modulus 3. Because N = 0 for fl = 0 and a1 0 mod 3 for *
*every
a = s1a1s2a22 Ker(N) for fl = 1, 2, we obtain the desired results.
o The argument to obtain all the p-centric subgroups isomorphic to Z=3 x Z*
*=3 is
similar. First, from the earlier discussion, they must be of the form ** for i as in
the statement and some a 2 fl1. As before, the condition for to hav*
*e order 3 is
N(a) = 0, and ** and ** are B(3, r; 0, fl, 0)-conjugate if an*
*d only if a1 and a01
are in the same class modulus 3.
Lemma A.16. Let P be a proper p-centric subgroup of B(3, r; 1, 0, 0) isomorphic*
* to Z=3n x
Z=3n or C(3, n) for some n. Then P is determined, up to conjugation, by the fol*
*lowing table:
________________________________________________________________
| Isomorphism |Subgroup (up| |
| | | |
| type |to conjuga-|Conditions |
| | | |
| |tion) | |
|________________|_____________|_________________________________ |
| Z=3k-1xZ=3k-1 |fl2 = |r = 2k. |
|________________|_____________|________________________________||||
| 1+2 | def 0 | k-1 k-2 |
| 3+ |E0 = **|i =s 3 , i0=s -3 for r =2k+1,|
|________________|______________| 2k-2 3k-2 |
| | def |i =s 3 , i0=s 3 for r =2k. |
| Z=3 x Z=3 |V0 = ** | 3 2 |
|________________|_____________|________________________________|
Proof. Consider the following short exact sequence induced by the abelian chara*
*cteristic sub-
group fl2 = of B(3, r; 1, 0, 0):
1 ! fl2 ! B(3, r; 1, 0, 0) i!Z=3 x Z=3 ~=<_s, __s1> ! 1.
34 ANTONIO D'IAZ, ALBERT RUIZ, AND ANTONIO VIRUEL
If P is contained in fl2 then, as fl2 is abelian and P must contains its centra*
*lizer, P must
equal fl2 = .
Recalling the orders of s2 and s3 we check that only the case r even is allow*
*ed.
Suppose now that P is not contained in fl2 and consider the non-trivial subgr*
*oup ß(P ):
Case ß(P ) = <__s1>,<_s_s1> or <_s-1_s1>: then P fits in a non-split short exac*
*t sequence:
1 ! K ! P !i Z=3 ! 1,
with K fl2. Checking cases for K and P we obtain that in any case this short*
* exact
sequence would split, which is a contradiction.
Case ß(P ) = <_s>: then P is a subgroup of ø = ~=B(3, r - 1; 0, 0, *
*0). Now apply
Lemma A.15 and notice that conjugation by s1 conjugates the three copies of 31+*
*2+and
Z=3 x Z=3.
Case ß(P ) = Z=3 x Z=3: If P 6= B(3, r; 1, 0, 0) then there exists a maximal pr*
*oper subgroup
H < B containing P . As fl2 is the Frattini subgroup of B(3, r; 1, 0, 0), that *
*is, the intersection
of the maximal subgroups, then ß(H) = Z=3. This is a contradiction with ß(P ) =*
* Z=3xZ=3,
and thus P equals B(3, r; 1, 0, 0).
In the proof of Theorem 5.9 we use implicitly some particular copies of SL2(3*
*) and GL 2(3)
lying in Aut(fl1). These are characterized by containing a fixed matrix. In the*
* next lemma
we show when they do exist:
Lemma A.17. Consider P ~=Z=3k x Z=3k and M2k+1,flsthe matrix defined in Remark *
*A.13
for the case B(3, 2k + 1; 0, fl, 0). Then:
o For fl = 0 there is, up to conjugacy, one copy of SL 2(3) (respectively *
*GL 2(3)) in
Aut(P ) containing M2k+1,0s.
o For fl = 1 there is, up to conjugacy, one copy of SL2(3) (respectively n*
*one of GL 2(3))
in Aut(P ) containing M2k+1,1s.
Proof.As SL2(3) and GL 2(3) are 3-reduced any copy of these groups lying in Aut*
*(P ) is a lift
j
of a subgroup of GL 2(3) by the Frattini map Aut(P ) ! GL 2(3).
It can be checked that for k = 2 the statements of the Lemma are true. Now, c*
*all P 0to
the Frattini subgroup of P , P 0def=Z=3k-1 x Z=3k-1, and consider the restricti*
*on map with
abelian kernel:
1 ! (Z=3)4 ! Aut(P ) i!Aut (P 0) ! 1.
Let A denote (Z=3)4. We use this exact sequence to prove the statements by indu*
*ction on k.
We suppose the lemma is true for k -1 and prove it for k 3. We take G = SL2(3*
*) or GL 2(3)
and fl = 0 or 1. Call Lk,fldef=M2k+1,fls, where M2k+1,flsis the matrix defined *
*in Remark A.13. It
is straightforward that ß(Lk,0) = Lk-1,0and ß(Lk,1) = Lk-1,0. We prove the lemm*
*a in three
steps:
Existence: We show the existence of the three stated copies. By hypothesis th*
*ere exists
a lift G !ffAut (P 0) such that æoe = IdG and with ~ def=Lk-1,0= 1-31-22 oe(G*
*). Write
p-LOCAL FINITE GROUPS OF RANK TWO FOR ODD p *
* 35
H def=<~> 2 Syl3(oe(G)) and take the pullback twice:
" i 0
A Ø_____//_Aut(P_)___////_Aut(P )
|| OO OO
|| | |
|| | |
||Ø " ?Ø| i ?Ø|
A ____//_ß-1(oe(G))__////_oe(G)
|| OO OO
|| | |
|| | |
||Ø " ?Ø| i ?Ø|
A _____//_ß-1(H)______////_H
As Lk,0= 1-31-2lies in ß-1(H) the bottom short exact sequence splits and its m*
*iddle term
can be identified with A : Lk,0. Notice also that Lk,1lies in A : Lk,0, there a*
*re several lifts
of H and the A-conjugacy classes of these lifts are in 1 - 1 correspondence wit*
*h H1(H; A).
In fact, each section H ! ß-1(H), corresponds, by the earlier identification, t*
*o a derivation
d: H ! A, such that d(~) = a~ and d(~2) = b~2 (notice that the ~'s inside and o*
*utside the d
lie in different automorphism groups).
Recall that we are interested in building lifts G ! Aut (P ) containing Lk,fl*
*, for which
is enough to give sections of the middle short exact sequence in the diagram ab*
*ove which
images contains Lk,fl. If this sequence splits then the A-conjugacy classes of *
*its sections are
in 1 - 1 correspondence with H1(G; A) (for clarity we do not write H1(oe(G); A)*
*), and the
sections which image contains Lk,flare precisely those which goes by the restri*
*ction map
resGH:H1(G; A) ! H1(H; A) to the class of the section induced by Lk,flin the bo*
*ttom short
exact sequence.
In fact, that the middle sequence splits is due to [G : H] being invertible i*
*n A and applying
a transfer argument we find that resGH:H*(G; A) ! H*(H; A) is a monomorphism (a*
*nd
corHG:H*(H; A) ! H*(G; A) is an epimorphism), so the class of the middle sequen*
*ce goes by
resGH:H2(G; A) ! H2(H; A) to the class of the bottom sequence, which is zero, a*
*nd must be
the zero class too, that is, the split one.
Finally, the sections oe0, oe1: H ! ß-1(H) which take ~ to Lk,0and Lk,1corres*
*pond to
the identically zero derivation and to a = 31k-110 3k-1, b = 101 03k-1respec*
*tively. These
sections are in the image of the restriction map resGH:H1(G; A) ! H1(H; A) if a*
*nd only if
they are in the G-invariants in H1(H; A). And easy check shows that z 2 H1(H; *
*A) is a
G-invariant if and only if gz = z for every g 2 Nff(G)(H). Once computed the d*
*erivations
Der (H, A) and the principal derivations P (H, A), we apply the action of g on *
*oe0 and oe1 at
the cochain level, and check that the class of oe0 is always G-invariant and th*
*at the class of oe1
is G-invariant just for G = SL2(3) in H1(H; A) ~=Der(H, A)=P (H, A).
Uniqueness: Now we show that the three found copies are unique up to Aut(P )-co*
*njugation,
as claimed. We use induction and the same tools as in the first step. Take two *
*lifts oe1, oe2 :
G ! Aut (P ) containing Lk,fl. Composing with ß we obtain two maps from G to A*
*ut (P 0)
containing Lk-1,0. They are lifts if they are injective, that is, if A\oei(G) i*
*s trivial for i = 1, 2.
As GL 2(3) and SL2(3) are 3-reduced, and as A is a normal 3-group, these groups*
* are indeed
trivial. So, by the induction hypothesis, the two lifts arriving at Aut(P 0) mu*
*st be conjugated
by some g02 Aut(P 0) which centralizes Lk-1,0.
36 ANTONIO D'IAZ, ALBERT RUIZ, AND ANTONIO VIRUEL
It is a straightforward calculation that the order of the centralizers CAut(P*
*)(Lk,fl) for fl = 0, 1
is 2 . 32k-1, and that of A \ CAut(P)(Lk,fl) for fl = 0, 1 is 9 (for every k *
*2). Because
ß maps CAut(P)(Lk,fl) to CAut(P0)(Lk-1,0), a element counting argument shows th*
*at in fact
ß(CAut(P)(Lk,fl)) = CAut(P0)(Lk-1,0).
Thus, there exists g 2 Aut(P ) with ß(g) = g0 and such that g centralizes Lk,*
*fl. Therefore
the images of oe1 and oe02def=cg O oe2 contain Lk,fland have the same image by *
*ß, that is, they
both lie in A : oe1(G) = ß-1(ßoe1(G)).
Choosing the Sylow 3-subgroup H = <~> of ßoe1(G) we can construct a three row*
*s short
exact sequences diagram as before, and argue using the injectivity of the restr*
*iction map
resGH:H1(G; A) ! H1(H; A) to obtain the uniqueness. More precisely, as the two *
*sections
ß-1 :ßoe1(G) ! oe1(G) and ß-1 : ßoe1(G) ! oe02(G) of the middle sequence of the*
* diagram
induce the same section in the bottom row, that is, the one which maps ~ to Lk,*
*fl, they must
be in the same class in H1(G; A), which means that they are A-conjugate. There*
*fore oe1
and oe2 are Aut(P )-conjugate.
Non existence: The arguments of the two preceding parts prove also the non exis*
*tence of
sections GL 2(3) ! Aut(P ) containing Lk,1.
Remark A.18. A cohomology-free proof of the non existence of copies (lifts) of *
*GL 2(3) in
Aut (Z=3k x Z=3k) containing Lk,1= M2k+1,1sruns as follows: if this were the ca*
*se, then, as
the elements of order 3 form a single conjugacy class in GL 2(3), we would obta*
*in that Lk,1
and its square are conjugate, and so would have same determinant and trace. But*
* one can
checks that this is not the case.
If H and K are two groups and H acts on K by ' : H ! Aut(K) then we can const*
*ruct
the semidirect product K :' H. In fact, if _ : H ! Aut (K) is another action c*
*onjugate
to ', that is, exists ff 2 Aut (K) such that _(h) = ff-1 O '(h) O ff for every *
*h 2 H, then
K :' H ~=K :_ H. The lemma above implies:
Corollary A.19. There exists the groups fl1 : SL2(3) and fl1 : GL 2(3) where th*
*e actions maps
100-1 to M2k+1,fl
s , with fl = 0, 1 for SL2(3) and fl = 0 for GL 2(3). Moreover, t*
*hese semidirect
products with actions as stated are unique up to isomorphism.
References
[1]A. Adem, J.F. Davis, O. Unlu, Fixity and Free Group Actions on Products of *
*Spheres, preprint.
[2]A. Adem, J. Smith, Periodic complexes and group actions, Ann. of Math. (2) *
*154 (2001), no. 2, 407-435.
[3]J.L. Alperin, Weights for finite groups, in The Arcata Conference on Repres*
*entations of Finite Groups
(Arcata, Calif., 1986), 369-379, Proc. Sympos. Pure Math., Part 1, Amer. Ma*
*th. Soc., Providence, RI,
1987.
[4]J.L. Alperin, R.B. Bell, Groups and Representations, Graduate Texts in Math*
*ematics, 162. Springer-
Verlag, New York-Berlin, 1995.
[5]J.L. Alperin, P. Fong, Weights for symmetric and general linear groups, J. *
*Algebra 31 (1990), 2-22.
[6]N. Blackburn, On a special class of p-groups, Acta Math. 100 (1958), 45-92.
[7]N. Blackburn, Generalizations of certain elementary theorems on p-groups, P*
*roc. London Math. Soc. 11
(1961), 1-22.
[8]K.S. Brown, Cohomology of groups, Graduate Texts in Mathematics, 87. Spring*
*er-Verlag, New York-
Berlin, 1982.
p-LOCAL FINITE GROUPS OF RANK TWO FOR ODD p *
* 37
[9]C. Broto, N. Castellana, J. Grodal, R. Levi, R. Oliver, Extensions of p-loc*
*al finite groups, preprint.
[10]C. Broto, R. Levi, R. Oliver, Homotopy equivalences of p-completed classify*
*ing spaces of finite groups,
Invent. Math. 151 (2003), 611-664.
[11]C. Broto, R. Levi, R. Oliver, The homotopy theory of fusion systems, J. Ame*
*r. Math. Soc. vol. 16 (2003),
no. 4, 779-856.
[12]C. Broto, R. Levi, R. Oliver, The theory of p-local groups: A survey, Conte*
*mp. Math. (to appear).
[13]C. Broto, J.M. Møller, Finite Chevalley versions of p-compact groups, prepr*
*int.
[14]W. Burnside, Theory of groups of finite order, Dover, New York, (1955).
[15]J. Conway, R. Curtis, S. Norton, R. Parker, R. Wilson, Atlas of finite grou*
*ps, Oxford Univ. Press,
London, (1985).
[16]B.N. Cooperstein, Maximal groups of G2(2n), Journal of Algebra 70 (1981), 2*
*3-36.
[17]E.C. Dade, Counting characters in blocks. II.9, in Representation theory of*
* finite groups (Columbus, OH,
1995), 45-59, de Gruyter, Berlin, 1997.
[18]J. Dietz, Stable splittings of classifying spaces of metacyclic p-groups, p*
* odd, J. Pure Appl. Algebra 90
(1993), 115-136.
[19]J. Dietz, S. Priddy, The stable homotopy type of rank two p-groups, Homotop*
*y theory and its applications
(Cocoyoc, 1993), 93-103, Contemp. Math., 188, Amer. Math. Soc., Providence,*
* RI, (1995).
[20]D. Gorenstein, Finite groups. Harper & Row, (1968).
[21]D. Gorenstein, Finite simple groups. An introduction to their classificatio*
*n. University Series in Mathe-
matics. Plenum Publishing Corp., New York, (1982).
[22]D. Gorenstein, The classification of finite simple groups. Vol. 1. Groups o*
*f noncharacteristic 2 type. The
University Series in Mathematics. Plenum Press, New York, (1983).
[23]D. Gorenstein, R. Lyons, The local structure of finite groups of characteri*
*stic 2 type, Mem. Amer. Math.
Soc. 42 (1983), no. 276.
[24]D. Gorenstein, R. Lyons, R. Solomon, The classification of the Finite Simpl*
*e Groups, Mathematical
Surveys and Monographs, 40.1 AMS (1994).
[25]D. Gorenstein, R. Lyons, R. Solomon, The classification of the finite simpl*
*e groups. Number 3. Part I.
Chapter A. Almost simple K-groups, Mathematical Surveys and Monographs, 40.*
*3. AMS (1998).
[26]B. Huppert, Endliche Gruppen, Springer, Berlin, Heidelberg, (1967).
[27]P.B. Kleidman, The maximal subgroups of the Steinberg triality groups 3D4(q*
*) and of their automorphism
groups, J. Algebra 115 (1988), 182-199.
[28]P.B. Kleidman, The maximal subgroups of Chevalley groups G2(q) with q odd, *
*the Ree groups 2G2(q),
and their automorphism groups, J. Algebra 117 (1988), 30-71.
[29]I. J. Leary, The Cohomology of Certain Groups, PhD Thesis, (1992).
[30]R. Levi, R. Oliver, Construction of 2-local finite groups of a type studied*
* by Solomon and Benson, Geom.
Topol. 6 (2002), 917-990.
[31]G. Malle, The maximal subgroups of 2F4(q2), J. Algebra 139 (1991), 52-69.
[32]J. Martino, S. Priddy, On the cohomology and homotopy of Swan groups. Math.*
* Z. 225 (1997), 277-288.
[33]R. Oliver, Equivalences of classifying spaces completed at odd primes, prep*
*rint.
[34]Ll. Puig, Structure locale dans les groupes finis, Bull. Soc. Math. France *
*Suppl. M'um. No. 47 (1976),
132 pp.
[35]A. Ruiz, A. Viruel, The classification of p-local finite groups over the ex*
*traspecial group of order p3 and
exponent p. Math. Zeit. (to appear).
[36]R. Stancu, Propri'et'ees locales des groupes finis, Th`ese de doctorat pr'e*
*sent'ee `a la Facult'e des Sciences de
l'Universit'e de Lausanne (2002).
[37]R. Stancu, Almost all generalized extraspecial p-groups are resistant, J. A*
*lgebra 249 (2002), 120-126.
38 ANTONIO D'IAZ, ALBERT RUIZ, AND ANTONIO VIRUEL
(A. D'iaz, A Viruel) Departamento de 'Algebra, Geometr'ia y Topolog'ia, Unive*
*rsidad de M'a-
laga, Apdo correos 59, 29080 M'alaga, Spain.
E-mail address, A. D'iaz: adiaz@agt.cie.uma.es
E-mail address, A. Viruel: viruel@agt.cie.uma.es
(A. Ruiz) Departament de Matem`atiques, Universitat Aut`onoma de Barcelona, 0*
*8193 Cer-
danyola del Vall`es, Spain.
E-mail address, A. Ruiz: Albert.Ruiz@uab.es
*