THE CLASSIFICATION OF p-LOCAL FINITE GROUPS OVER THE
EXTRASPECIAL GROUP OF ORDER p3 AND EXPONENT p
ALBERT RUIZ AND ANTONIO VIRUEL
1.Introduction
The concept of p-local finite group arise in the work of Broto-Levi-Oliver [2*
*] as a
generalization of the classical concept of finite group. These purely algebraic*
* objects,
defined in Section 2, are the culmination of a research programme, carried by s*
*everal
authors over the last three decades, in two apparently unrelated areas: Group T*
*heory
and Homotopy Theory.
Back in the 70's, and influenced by the celebrated work of Alperin, Puig [11]*
* set out
a systematic framework for the study of a finite group G in terms of what is kn*
*own as
the p-local structure of G or its p-fusion: its p-subgroups, their normalizers,*
* and the
relations given by conjugation in G. This axiomatic development of p-fusion sys*
*tem,
reviewed in Section 2, has been proved to be an useful tool for determine many *
*of
the properties of G and the p-completion of its classifying space BG (see for e*
*xample
[9]). Unfortunately, p-fusion is not enough to determine all the homotopy theor*
*etic
information related to BG^pas the results in [1] show. The complete description*
* of
BG^prequires the use of an additional structure, the centric linking system, an*
*d leads
to the definition of p-local finite group (Definition 2.7).
The theory of p-local finite groups is a generalization of the classical theo*
*ry of finite
groups in the sense that every finite group leads to a p-local finite group, bu*
*t there
exist exotic p-local finite groups which are not associated to any finite group*
* as it
can be read in [2, Sect. 9], [8], or Lemma 4.16. Therefore, the classification*
* of p-
local finite groups has interest, not only by itself but, as an opportunity to *
*enlighten
one of the highest mathematical achievements in the last decades: The Classific*
*ation
of Finite Simple Groups [6]. This Classification of Finite Simple Groups provid*
*es 26
mathematical gems, 26 sporadic finite simple groups that enjoy an intriguing pr*
*operty:
if G is an sporadic finite simple group with p-Sylow S G, p > 2, of order p3,*
* then S
is isomorphic to the extraspecial group of order p3 and exponent p, denoted by *
*p1+2+,
and p 13. This fact, partially explained in Corollary 4.11, is the start poin*
*t for the
classification of p-local finite group over the p-groups of type p1+2+.
___________
2000 Mathematics subject classification 55R35, 20D20
First author is partially supported by MCYT grant BFM2001-2035.
Second author is partially supported by MCYT grant BFM2001-1825.
Both authors have been supported by the EU grant nr. HPRN-CT-1999-00119.
1
2 ALBERT RUIZ AND ANTONIO VIRUEL
Theorem 1.1. Let p be an odd prime. If (p1+2+, F, L) is a p-local finite group *
*then it
is completely determined by the saturated fusion system (p1+2+, F) and it corre*
*sponds
to one in the following tables:
____________________________________________________________________________
||| Out F(p1+2+) ]Fec-rad AutF(V ) Group Condition |
|||__________________________________________________________________________|
||| W 0 ; p1+2 : W p - ||W | |
|||___________________________________________________+___________|________|_
||| (p - 1) x r 1 SL 2(p).r p2:(SL 2(p).r)|r|(p - 1)*
* |
|||_______________________________________________________________|_________ *
* |
||| (p-1)_ (p-1)_ | *
* |
||| (p - 1) x 3 1 + 1 SL 2(p) : 3 L3(p) | *
* |
|||_______________________________________________________________(p-1)(p-1)|*
* |
||| (p - 1) x ____ : 2 2 SL 2(p) : ____ L3(p) : 2 | *
* |
|||_____________3_______________________________3_________________|3|(p - 1) *
* |
||| 2 | *
* |
||| (p - 1) 1 + 1 GL 2(p) L3(p) : 3 | *
* |
|||_______________________________________________________________| *
* |
||| (p - 1)2 : 2 2 GL 2(p) L3(p) : S3 | *
* |
|||_______________________________________________________________|_________ *
* |
||| 2 | *
* |
||| (p - 1) 1 + 1 GL 2(p) L3(p) | *
* |
|||_______________________________________________________________|3 - (p - 1*
*) |
||| (p - 1)2 : 2 2 GL 2(p) L3(p) : 2 | *
* |
|||_______________________________________________________________|_________ *
* |
Table 1. Semidirect products and extensions of linear groups
____________________________________________________________________________
||| Out F(p1+2+) ]Fec-rad AutF(V ) Group Condition |
|||__________________________________________________________________________|
||| 2 0 | *
* |
||| D8 2 + 2 GL 2(p) F4(2) | *
* |
|||_______________________________________________________________|p = 3 *
* |
||| SD16 4 GL 2(p) J4 | *
* |
|||_______________________________________________________________|_________ *
* |
||| 4S4 6 GL 2(p) T h p = |5 |
|||_______________________________________________________________|_______|_|*
*|||||
||| S3 x 3 3 SL 2(p) He | |
|||_______________________________________________________________| |
||| | |
||| S3 x 6 3 SL 2(p) : 2 He : 2 | |
|||_______________________________________________________________| |
||| S3 x 6 3 + 3 SL 2(p) : 2 F i0 | |
|||_____________________________________________________24________| |
||| 2 | |
||| 6 : 2 6 SL 2(p) : 2 F i24 | |
|||_______________________________________________________________| |
||| 62 : 2 6 + 2 SL 2(p):2, GL 2(p) p = |7 |
|||_______________________________________________________________| |
||| 0 | |
||| D8 x 3 2 + 2 SL 2(p) : 2 O N | |
|||_______________________________________________________________| |
||| D16x 3 4 SL 2(p) : 2 O0N : 2 | |
|||_______________________________________________________________| |
||| | |
||| D16x 3 4+4 SL 2(p) : 2 | |
|||_______________________________________________________________| |
||| SD32x 3 8 SL 2(p) : 2 | |
|||_______________________________________________________________|_______|_
||| 3 x 4S4 6 SL 2(p).4 M p = |13 |
|||_______________________________________________________________|_______|_
Table 2. Sporadic groups and exotic p-local finite groups
where Fec-radis the set of elementary abelian F-centric and F-radical p-subgrou*
*ps,
its cardinal is separated by F-conjugation classes, and Aut F(V ) is the group *
*of F-
automorphisms for each representative V of the F-conjugacy classes in Fec-rad.
p-LOCAL FINITE GROUPS WITH p-SYLOW p1+2+ 3
Proof.See Section 4.
Remark 1.2. Notice that the classification above provides three new examples of
7-local finite groups. These are the simplest examples of exotic p-local finite*
* groups
known so far ([2, Section 9], [8]) and are intensively studied in a sequel pape*
*r [12]. The
proof of the nonexistence of a finite group whose 7-fusion data is isomorphic t*
*o any of
those examples (Lemma 4.16) is heavily based in the Classification of Simple Gr*
*oups
[6]. Therefore, within our philosophy, a more conceptual proof would be desirab*
*le.
Remark 1.3. Notice also that the groups in both tables are not the only ones wi*
*th
these fusion systems, even if we consider just simple finite groups. The follow*
*ing table
completes the list of sporadic simple groups, whose p-Sylow is also of the form*
* p1+2+,
and their corresponding fusion system.
_____________________________________________
| Out F(p1+2+)]Fec-rad AutF (V ) Group p |
|____________________________________________|_
| 22 1 + 1 GL 2(3) M12 3 |
|____________________________________________|
| 8 0 ; J2 3 |
|____________________________________________|
| D8 2 GL 2(3) M24, He 3 |
|____________________________________________|
| SD16 4 GL 2(3) Ru 3 |
|____________________________________________|
| 42 : 2 2 GL 2(5) Ru 5 |
|____________________________________________|
| 24 : 2 0 ; Co3 5 |
|____________________________________________|
| 4S4 0 ; Co2 5 |
|____________________________________________|
| 8 : 2 0 ; HS 5 |
|____________________________________________|
| 3 : 8 0 ; McL 5 |
|____________________________________________|
| 5 x 2S4 0 ; J4 11 |
|____________________________________________|
Finally, Theorem 1.1 is all we need to classify all p-local finite groups (p-*
*odd) over
p-groups of order p3:
Corollary 1.4. Let (S, F, L) be a p-local finite group, p > 2, such that |S| = *
*p3.
Then (S, F, L) is completely determined by the saturated fusion system (S, F) a*
*nd it
corresponds to one of the following:
o The fusion system of the group S : AutF (S) if S 6~=p1+2+.
o One of the fusion systems in the tables in Theorem 1.1 if S ~=p1+2+.
Proof.If |S| = p3, then S is either abelian, or generalized extraspecial [13, D*
*efinition
3.1]. Therefore, if S 6~= p1+2+, then S is a resistant group [13, Theorem 4.2]*
*, that
is, F = NAutF(S)F(S) [2, Definition A.3] and F is the saturated fusion system o*
*f the
group S : AutF (S). Finally, the obstruction classes to the existence and uniqu*
*eness
of centric linking systems associated to the saturated fusion system of a group*
* of type
S : W , where p - |W |, live in H*(Z(S); W ) = 0. If S ~= p1+2+we apply Theorem
1.1.
4 ALBERT RUIZ AND ANTONIO VIRUEL
Organization of the paper: In Section 2, we briefly review the general theory of
p-local finite groups. In Section 3, we describe the basic properties of the gr*
*oup p1+2+
that are going to be used along of the last section, where the proof of Theorem*
* 1.1 is
worked out.
Notation: By p we always denote an odd prime. The group theoretical notation
used along this paper is that described in the Atlas [4, 5.2]. For a group G, *
*and
g 2 G, we denote by cg the conjugation morphism x 7! gxg-1. If P, Q G, the se*
*t 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 ). Notice that AutG (G) is then the set of inner automorph*
*isms
of G.
Acknowledges: The authors would like to thank Bob Oliver his many helpful com-
ments and suggestions throughout the development of this work.
2. p-local finite groups
Along this section, we quickly review the basic definitions and results rela*
*ted to
the theory of p-local finite groups we shall use. We refer to [2] and [3] for *
*a more
complete description and properties of these objects.
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 follo*
*wing
two conditions:
(a) Hom S(P, Q) Hom F(P, Q) Inj(P, Q) for all P , Q subgroups of S, where
Hom G means the morphisms induced by conjugation in G and Injthe injective
morphisms.
(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, and by condition (b) we *
*denote
by AutF (P, Q) the morphisms in F from P to Q when |P | = |Q| and by AutF (P ) *
*the
group AutF (P, P ). We denote by Out F(P ) the quotient group AutF (P )= AutP(P*
* ).
The fusion systems that we will consider satisfy the following conditions:
Definition 2.2. Let F 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 *
*all P 0
which is F-conjugate to P .
o A subgroup P S is fully normalized in F if |NS(P )| |NS(P 0)| for a*
*ll P 0
which is F-conjugate to P .
o F is a saturated fusion system if the following two conditions hold:
(I)Each fully normalized subgroup P S is fully centralized and AutS(*
*P ) 2
Sylp(Aut F(P )).
(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 = '.
p-LOCAL FINITE GROUPS WITH p-SYLOW p1+2+ 5
Remark 2.3. From the definition of fully normalized and the condition (I) of sa*
*tu-
rated fusion system we get that if F is a saturated fusion system over a p-grou*
*p S
then p cannot divide the outer automorphism group Out F(S).
As expected, every finite group G gives rise to a saturated fusion system [2,*
* Propo-
sition 1.3], which provides valuable information about BG^p[9]. Some classical*
* re-
sults for finite groups can be generalized to saturated fusion systems, as for *
*example,
Alperin's fusion theorem [2, Theorem A.10]:
Definition 2.4. Let F be any fusion system over a p-group S. A subgroup P S
is:
o F-centric if P and all its F-conjugates contain their S-centralizers. L*
*et Fc
denote the full subcategory of F whose objects are the F-centric subgrou*
*ps
of S.
o F-radical if Out F(P ) is p-reduced, that is, if Out F(P ) has no proper*
* normal
p-subgroup.
Theorem 2.5. 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 normalizer 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-1 O . .O._1.
Nevertheless, the definition of p-local finite group still needs of some extr*
*a structure:
Definition 2.6. 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 subgroup*
*s 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, Q 2 L, Z(P ) acts freely on Mor L(P, Q) by compo*
*sition
(upon identifying Z(P ) with ffiP(Z(P )) AutL(P )0, and ß induces a bi*
*jection
~=
MorL(P, Q)=Z(P )- ! Hom F (P, Q)
(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:
6 ALBERT RUIZ AND ANTONIO VIRUEL
Definition 2.7. 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 associate*
*d 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 theo*
*ry for
the existence and uniqueness of a centric linking system, i.e. p-local finite *
*groups,
associated to F. The question is solved for p-groups of small rank by the follo*
*wing
result [2, Theorem E]:
Theorem 2.8. Let F be any saturated fusion system over a p-group S. If rk(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.
All p-local finite groups studied in this work are over the p-group p1+2+. *
*As the
p-rank of p1+2+is 2, we obtain:
Corollary 2.9. Let p be an odd prime. Then the set of p-local finite groups ove*
*r the
p-group p1+2+is in bijective correspondence with the set of saturated fusion sy*
*stems
over p1+2+.
3.The group p1+2+
In this section, we collect the basic properties of p1+2+, the extraspecial *
*group of
order p3 and exponent p (p odd), that we shall use along the paper.
The group p1+2+is usually presented as
p1+2+~=
so p1+2+is generated by the elements A and B, and C appears as a generator of
Z(p1+2+) = [p1+2+, p1+2+]. Therefore any automorphism of p1+2+is determined by*
* the
images of the elements A and B. We can read in [7], that the automorphisms of p*
*1+2+
0 s0 t0 r s t
are indeed those endomorphisms which map A to Ar B C and B to0A B C . The
inner automorphisms are then automorphisms of the form A 7! ACt and B 7! BCt,
so we can describe the outer automorphisms of p1+2+as:
Lemma 3.1. The outer automorphism group of p1+2+is isomorphic to GL 2(p): the
matrix r0rs0swith determinant j corresponds to the class of the automorphism w*
*hich
0 s0 r s j
sends A to Ar B , sends B to A B and C to C .
The following Lemma gives us the structure of the F-centric subgroups of p1+*
*2+,
which is a fundamental tool in the study of the possible saturated fusion syste*
*ms.
Lemma 3.2.
(a) There are exactly (p + 1) rank two elementary abelian subgroups in p1+2+tha*
*t we
label as Vi= for i = 0, . .,.(p - 1) and Vp = .
(b) Every inner automorphism of p1+2+restricts to an automorphism of Vi for all*
* i.
(c)Assume F is a fusion system over p1+2+. Then the F-centric subgroups are t*
*he
total and the rank two elementary abelian subgroups.
p-LOCAL FINITE GROUPS WITH p-SYLOW p1+2+ 7
Proof.Statements (a) and (b) follow from the presentation of p1+2+and the descr*
*iption
of its automorphisms given above.
In order to obtain (c), recall that P is F-centric if and only if P and all i*
*ts F-
conjugates contain their p1+2+-centralizers. Let P be F-centric. As = Z(p*
*1+2+),
then fi P . If we add just one element to , we will have a rank two elem*
*entary
abelian self-centralizing p-subgroup. As the F-conjugates of a rank two element*
*ary
abelian p-subgroup must be again a rank two elementary abelian p-subgroup, then
it will be also self-centralizing and that means that the rank two elementary a*
*belian
p-subgroups are the smaller F-centric subgroups.
If we add any element to a rank two elementary abelian p-subgroup, then we wi*
*ll
have the total, so the result follows.
4.Proof of the classification
The aim of this section is to provide a proof of Theorem 1.1. This proof appe*
*ars
at the end of the section and is subdivided in the following series of lemmas. *
* The
notation used for the elements and subgroups of p1+2+is the one described in the
previous section, and F will always denote a saturated fusion system over p1+2+.
According with Alperin's theorem for fusion systems, Theorem 2.5, the fusion *
*sys-
tem F is determined by the full subcategory of F-centric and F-radical subgroup*
*s.
In Lemma 3.2 we show that p1+2+and its rank two elementary abelian p-subgroups
are all F-centric. As p1+2+is always F-radical, it remains to see which of thes*
*e rank
two elementary abelian p-subgroups can be F-radical. Let Fec-raddenote the set *
*of
elementary abelian F-centric and F-radical p-subgroups, that is, the F-radical *
*rank
two elementary abelian p-subgroups.
We begin with a characterization of the elements in Fec-rad.
Lemma 4.1. Let V be a rank two elementary abelian p-subgroup of p1+2+. Then
V 2 Fec-radif and only if SL2(p) AutF (V ).
Proof.If SL2(p) AutF (V ) = Out F(V ) GL 2(p), then Aut F(V ) is p-reduced *
*and
V is F-radical. So V 2 Fec-rad
Now, let V be in Fec-rad. As it is a rank two elementary abelian p-subgroup*
*, it
must be one of the Vi's, generated by C and X = AiBj with (i, j) 6= (0, 0). We
think of {C, X} as a Fp-basis of V . In this fixed basis, the matrix m = 1011,*
* which
generates a subgroup of order p, is in AutF (V ), because is the restriction of*
* an inner
automorphism of p1+2+. As V is abelian, hence Out F(V ) = AutF (V ), and F-radi*
*cal,
then there exist matrices m02 OutF (V ) conjugated to m which are not in .
We have that Aut F(V ) acts on the projective space L of lines in F2p, and th*
*at
divides L in two orbits, one of them with the single element (1, 0). Consi*
*der
an element m0 2 Out F(V ) conjugated to m, which will take the element (1, 0) to
another, making the action of transitive on L. Then there exists n 2
which maps (1, 0) 7! (1, 1). As m, m02 SL2(p), then n 2 SL2(p), and n is a matr*
*ix
of the form: ` '
1 x
1 x + 1 .
8 ALBERT RUIZ AND ANTONIO VIRUEL
Finally, nm-x = 11012 AutF (V ). But 1101and 1011generate SL2(p*
*),
hence SL2(p) AutF (V )
Fix a a primitive (p - 1) root of unity in F*p, and denote by (p - 1)-2 the *
*subgroup
generated by the matrix ` '
a 0
0 a-2 .
The next result shows how the number of subgroups of Out F(p1+2+) which are in *
*the
GL 2(p)-conjugacy class of (p - 1)-2 provides an upper bound for ]Fec-rad.
Lemma 4.2. For each V 2Fec-radthere is a different subgroup (p-1)V OutF (p1+2*
*+)
in the GL 2(p)-conjugacy class of (p - 1)-2.
Moreover (p-1)V breaks the set of rank two elementary abelian p-subgroups of*
* p1+2+
in the following orbits: two of one element, corresponding to the eigenvectors *
*of the
generator of (p - 1)V and:
(a) one of (p - 1) elements if 3 - (p - 1),
(b) three orbits of (p - 1)=3 elements otherwise.
Proof. Let V be in Fec-rad, with Fp basis {C, X} as in the proof of Lemma 4.1. *
*Let
a be the primitive (p - 1) root of the unity fixed above.
As SL2(p) Aut F(V ), then the map ' corresponding to the matrix a-100awill
be in Aut F(V ), and as all the rank two elementary abelian p-subgroups are ful*
*ly
centralized, it will extend to a map in N' (Definition 2.2), which is all p1+2+*
*. Consider
now this extension in Out F(p1+2+) GL 2(p), so we can think of it as a matrix*
* with
X as eigenvector of eigenvalue a and determinant a-1, so it has a-2 as the other
eigenvalue, different from a. We see that just one matrix with this properties *
*can be in
Out F (p1+2+) GL 2(p): if we had two different matrices such that X is an eig*
*envector
of eigenvalue a and determinant a-2, we can conjugate one of them to be diagonal
M1 = a00a-2, and by the properties assumed, the other must be M2 = a0ba-2,
with b 6= 0. Now we have that M-11M2 has order p, getting that Out F(p1+2+) has*
* an
element of order p, what cannot be true by Remark 2.3. With this the result abo*
*ut
the existence of as many different subgroups conjugated to (p - 1)-2 as F-radic*
*al
rank two elementary abelian p-groups follows.
Finally we take directly the subgroup (p - 1)-2 to check the action of its g*
*enerator
on the set {Vi}i=0..p, equivalent to the projective space of lines in F2p, and *
*we obtain
the invariant spaces of the statement.
Next lemma shows how AutF (p1+2+) and Fec-raddetermine AutF (V ) for every r*
*ank
two elementary abelian subgroup V p1+2+.
Lemma 4.3. Let V be a rank two elementary abelian subgroup of p1+2+. Then the
group Aut F(V ) is determined by Aut F(p1+2+) and the property of being or not *
*F-
radical.
Proof. If V is not F-radical, by Theorem 2.5, we get that the elements in Aut F*
*(V )
are restrictions of elements in AutF (p1+2+).
p-LOCAL FINITE GROUPS WITH p-SYLOW p1+2+ 9
Assume now that V is F-radical. We have that SL 2(p) Aut F(V ) GL 2(p),
so there exists r, divisor of (p - 1), such that Aut F(V ) ~= SL2(p).r. To see*
* that r
depends only on AutF (p1+2+) consider the elements in AutF (p1+2+) which restri*
*ct to an
element in AutF (V ). The image by the determinant of this restriction is an id*
*eal I0
generated by r0in Z=(p - 1). By definition r is one generator of the ideal I ge*
*nerated
by the image of the determinant applied to Aut F(V ) in Z=(p - 1). We claim th*
*at
these two ideals are the same: we have I0 I, so we have just to prove I I0. *
*Given
r002 I, and using that SL2(p) AutF (V ), we have that there are diagonal matr*
*ices
in Aut F(V ) with determinant r00. These diagonal matrices extend to an element*
* in
AutF (p1+2+) (we are using the same argument as in the proof of Lemma 4.2 about*
* the
extensions of maps to N'), so r002 I0, and I = I0, that is, r can be deduced fr*
*om
AutF (p1+2+).
A consequence of the previous lemma is:
Corollary 4.4. The category F is characterized by Out F(p1+2+) and the set Fec-*
*rad.
Proof.First observe that AutF (p1+2+) = Autp1+2+(p1+2+) : OutF (p1+2+). Thus Ou*
*tF (p1+2+)
determines AutF (p1+2+). By Lemma 4.3, AutF (p1+2+) and Fec-raddetermine AutF (*
*V )
for all V 2 Fec-rad. But, according to Alperin's theorem for fusion systems, Th*
*eorem
2.5, that is all the information we need to reconstruct the category F.
Therefore, in accordance with the corollary above, we proceed to classify the*
* pos-
sible F in terms of the possible combinations of OutF (p1+2+) and Fec-rad. The *
*easiest
case is:
Lemma 4.5. If Fec-rad= ;, then F is the fusion system of the group p1+2+: W ,
where W is Out F(p1+2+) GL 2(p) of order coprime to p.
Proof.Using Corollary 4.4, Out F(p1+2+) and the fact that Fec-rad= ; give all t*
*he
information and determines F.
The fact that p cannot divide the order of W follows from Remark 2.3.
Fix now W GL 2(p) of order coprime to p and consider the group G def=p1+2+:
W . Let F be the saturated fusion system associated to G. We have to check th*
*at
Fec-rad= ;. But, if V 2 Fec-rad, then SL2(p) AutF (V ), by Lemma 4.1, and that
is impossible as Z(p1+2+) < V is invariant by G-conjugation.
We next study the case of ]Fec-rad= 1.
Lemma 4.6. If ]Fec-rad= 1 then F is the fusion system of the group p2 : (SL 2(p*
*).r)
where SL 2(p).r GL 2(p) is the normal subgroup of index (p - 1)=r which conta*
*ins
SL2(p).
Proof.Assume that ]Fec-rad = 1. We can assume the only element in Fec-radis
V0 = and therefore (p - 1)-2 OutF (p1+2+).
As, V0 cannot be conjugated to any other rank two p-local finite subgroup, th*
*en
OutF (p1+2+) must be contained in the upper triangular matrices, but if this su*
*bgroup
contains any non-diagonal matrix, as it also contains (p-1)-2, then it will con*
*tain an
10 ALBERT RUIZ AND ANTONIO VIRUEL
element or order p. This implies that OutF (p1+2+) must be contained in the sub*
*group
of diagonal matrices which is of type (p - 1)2.
Then (p - 1)-2 OutF (p1+2+) (p - 1)2, and it follows that for each r, di*
*visor of
(p - 1), we have one subgroup isomorphic to (p - 1) x r and that these are all *
*the
possibilities for Out F(p1+2+).
Using the Corollary 4.4 we have that the category F is characterized by OutF*
* (p1+2+)
and the fact that V0 is the only radical.
Consider now the fusion system of the group G def=p2 : SL 2(p).r. It's clea*
*r that,
choosing V0 = p2 : 1 we obtain that AutF (V0) = SL2(p).r, Out F(p1+2+) = (p - 1*
*) x r,
and that there are no more F-radical rank two elementary abelian p-subgroups.
The study of the case ]Fec-rad = 2 requires a more complicated analysis that
depends on the existence and uniqueness (up to conjugation) of some subgroups of
GL 2(p).
Lemma 4.7.
(a) Up to conjugation, there is just one subgroup isomorphic to (p-1)2 : 2 in G*
*L 2(p).
(b) If Out F(p1+2+) = (p - 1)2 : 2 and p 6= 7, then ]Fec-rad= 0 or 2.
Proof. To prove (a) we think of (p - 1)2 < (p - 1)2 : 2 as a rank two abelian g*
*roup
generated by two matrices x and y. Any element of order (p - 1) is conjugated t*
*o a
diagonal matrix, so we can assume that x is a diagonal matrix. If y is also dia*
*gonal,
we have finish, using that the diagonal matrices in GL 2(p), as a group, are is*
*omorphic
to (p - 1)2. If y is not diagonal, using that x is diagonal and y must commute *
*with
x, we get that x is of the form a Id, and as x has order (p - 1), then a is a p*
*rimitive
(p - 1) root of the unity and is the center of GL 2(p), so we can conjugate*
* y to a
diagonal matrix without changing x and we get that, again, it is conjugated to *
*the
subgroup of diagonal matrices.
Assuming now that (p - 1)2 (p - 1) : 2 is the subgroup of diagonal matrice*
*s,
we need a non-diagonal matrix z of order 2 in NGL2(p)((p - 1)2), and we get that
z = 0110. Therefore there is just one possibility, up to conjugation, for a *
*group of
the form (p - 1) : 2.
To prove (b), we think of Out (p1+2+) = (p - 1)2 : 2 as the group of the dia*
*gonal
matrices and the twisting 0110. We can see that this group breaks the rank two*
* ele-
mentary abelian p-subgroups of p1+2+in two F-conjugacy classes, one with 2 elem*
*ents
(V0 and Vp) and another with (p - 1) elements (Vi for i = 1, . .,.p - 1).
If Fec-rad= {V0, Vp}, then Out FV0 = OutF Vp = GL 2(p) by Lemma 4.3, and F is
the fusion system of L3(p) : S3 if 3|(p - 1) and L3(p) : 2 otherwise.
Finally, if Fec-rad = {V1, . .,.Vp-1}, then it would imply that conjugating *
*by
the matrix -1111 the group Out (p1+2+) we should obtain upper triangular matri*
*ces
conjugated to (p-1)-2, corresponding to (p-1)Vp-1and we can check that we get j*
*ust
matrices of the form ~ Idand ~00-~, which generates one (p - 1)-2 only if ~3 =*
* -1
and then p must be 7.
Finally a classification for the case ]Fec-rad= 2.
p-LOCAL FINITE GROUPS WITH p-SYLOW p1+2+ 11
Lemma 4.8. If ]Fec-rad= 2 then F is one of the following saturated fusion syste*
*ms:
________________________________________________________________________
| Out F(p1+2+) ]Fec-rad AutF(V ) Group Condition |
|________________________________________________________________________|
| (p-1)_ (p-1)_ | |
| (p - 1) x 3 1 + 1 SL 2(p) : 3 L3(p) | |
|_____________________________________________________________(p-1)(p-1)| |
| (p - 1) x ____ : 2 2 SL 2(p) : ____ L3(p) : 2 | |
|_____________3_______________________________3_______________|3|(p - 1) |
| 2 | |
| (p - 1) 1 + 1 GL 2(p) L3(p) : 3 | |
|_____________________________________________________________| |
| (p - 1)2 : 2 2 GL 2(p) L3(p) : S3 | |
|_____________________________________________________________|_________ |
| 2 | |
| (p - 1) 1 + 1 GL 2(p) L3(p) | |
|_____________________________________________________________|3 - (p - 1) |
| (p - 1)2 : 2 2 GL 2(p) L3(p) : 2 | |
|_____________________________________________________________|_________ |
where ]Fec-rad is the cardinal of ]Fec-rad separated by F-conjugacy classes and
AutF (V ) is the group of F-automorphisms for each representative of the F-conj*
*ugacy
classes in ]Fec-rad.
Proof.Assume that Fec-rad= {V, V 0}, and let w and w0be matrices in OutF (p1+2+)
GL 2(p) corresponding to the generators of the groups (p - 1)V and (p - 1)V 0de*
*fined
in Lemma 4.2 We see that w and w0 have the same the eigenvectors (otherwise,
the F-class distribution given by Lemma 4.2 would imply ]Fec-rad > 2) so in an
appropriate basis we get that Out F(p1+2+) is contained in the subgroup generat*
*ed by
diagonal matrices and 0110 (again, any other non diagonal matrix would imply an
F-class distribution such that ]Fec-rad> 2).
The difference between being OutF (p1+2+) included in the diagonal matrices o*
*r not
will be that the two F-radical will appear in just one conjugation class (denot*
*ed as 2
in the table) or in two (denoted as 1 + 1 in the table).
Assume first that Out F(p1+2+) is included in the group of diagonal matrices,*
* as
]Fec-rad= 2 then Out F(p1+2+) must contain the subgroup generated by:
` ' ` '
a 0 a-2 0
0 a-2 and 0 a .
This subgroup generates all the diagonal matrices if 3 - (p - 1) and a subgroup
isomorph to (p - 1) x (p-1)_3if 3|(p - 1). In this second case we can also cons*
*ider all
the group (p - 1)2, getting the two different cases of the table.
Corollary 4.4 tells us that the category F is characterized by this data, and*
* Lemma
4.3 tells us how to compute the column AutF (V ) in the table.
Now, if 0110 2 Aut F(p1+2+), the same arguments work, getting the information
with the 2 in the table.
Finally we have to check that these fusion systems correspond to the groups i*
*n the
table. Consider S the p-Sylow of L3(p) as the represented by the classes of the*
* upper
triangular matrices with 1 in the diagonal. If we regard the conjugations in S*
* by
elements in L3(p) as a subgroup of GL 2(p), we obtain that these are the matric*
*es of
the form a00ab3, where a and b are invertible elements in Fp, so the group Out*
*F (p1+2+)
is (p - 1) x (p-1)_3or (p - 1)2 depending whether 3 divides (p - 1) or not.
12 ALBERT RUIZ AND ANTONIO VIRUEL
To check that ]Fec-rad= 2 consider elementary abelian rank two p-subgroup as*
* the
classes of matrices in the p-Sylow above with a 0 in the position (2, 3) and re*
*spectively
the other p-subgroup with a zero in the position (2, 1). Using the symmetry of *
*this
two subgroupsiwejhave just to checkithatjone of this is F-radical. To see thes*
*e,
100 100 1 -1 1 0
check that 010011(respectively 011001) acts as (0 1 )(respectively (-1 1)) *
*so, as
they generate SL2(p), these are F-radical. Moreover, applying the Lemma 4.7, ou*
*t of
p = 7 we cannot have more than 2 F-radical, and for p = 7 we can check the Atlas
[4] information about L3(7).
To complete the other cases just add that the action of the order two elemen*
*t in
Out (L3(p)) can be regarded as the matrix 0110in OutF (p1+2+) and, if 3 is a d*
*ivisor of
(p-1), we have also a cyclic group of order 3 in Out(L3(p)), which makes OutF (*
*p1+2+)
to be (p - 1)2.
Lemma 4.7 suggest that the analysis of the case p = 7 cannot fit in a genera*
*l frame-
work. The next results identify the other anomalous primes that we have to che*
*ck
case by case. We need the following results concerning the subgroups of PGL 2(p*
*).
Lemma 4.9. Let p an odd prime. The subgroups of PGL 2(p) of order coprime to p
are isomorphic to one of the following groups:
(a) A subgroup of the dihedral groups of order (p2 1).
(b) A4, S4.
(c)A5 if and only if 5 | p(p2 - 1).
Proof. Theorems 3.6.25 and 3.6.26 in [14] describe the possible subgroups of PS*
*L 2(q)
for q = pn. Then the result follows as a consequence of the inclusion of PGL 2(*
*p) in
PSL 2(p2) given in [14, 3.6.26(v)], and the fact that A5 is simple.
The relation about anomalous primes and the subgroups described in the previ*
*ous
lemma appears clear in the next result.
Lemma 4.10. Assume ]Fec-rad> 0 and denote by W~ the projection in PGL 2(p) of
Out F (p1+2+) GL 2(p). Then:
(a) If p > 7 then W~ cannot be in Dp2 1.
(b) If p > 5 and 3 - (p - 1) then W~ cannot be neither in A4, nor in S4, nor in*
* A5.
(c)If p > 16 and 3|(p - 1) then W~ cannot be neither in A4, nor in S4, nor in *
*A5.
Proof. Assume that {V, V 0, V 0} Fec-rad. Then the projection in PGL 2(p) of*
* the
groups (p-1)V , (p-1)V 0and (p-1)V 0will generate, at least, two different subg*
*roups
of order (p-1)_3or (p - 1), depending if 3|(p - 1) or not. Such situation can *
*happen
in Dp2 1if and only if either (p-1)_3or (p - 1) equals 2, that is, when p = 3 o*
*r p = 7,
what proves (a). Notice also that the order of the elements in the groups A4, S*
*4 and
A5 is less or equal than 5, hence:
(i) if 3 - (p - 1), then p - 1 5 what proves (b), and
(ii)if 3|(p - 1), then p-1_3 5 what proves (c).
p-LOCAL FINITE GROUPS WITH p-SYLOW p1+2+ 13
We can now give a partial answer, but independent of [6], to the question of *
*why
sporadic finite simple groups with p-Sylow of order p3, p > 2, have p-Sylow of *
*type
p1+2+and occur only for p 13.
Corollary 4.11. Let p be an odd prime and let (S, F, L) be a p-local finite gro*
*up over
S, a finite p group of order p3. If either S 6~= p1+2+, or p 6= 3, 5, 7, 13, th*
*en the p-local
finite group (S, F, L) is completely determined by the saturated fusion system *
*(S, F),
which is also the saturated fusion system associated to a group of type either
o S : W , where W Aut(S) and p - |W |,
o p2 : W , where SL2(p) W GL 2(p), or
o L3(p) : W , where W Out(L3(p)) and p - |W |.
Proof.The argument is the one used to prove Corollary 1.4. If |S| = p3, then S
is either abelian, or generalized extraspecial [13, Definition 3.1]. Therefore*
*, if S 6~=
p1+2+, then S is a resistant group [13, Theorem 4.2], that is, F = NAutF(S)F(S)*
* [2,
Definition A.3] and F is the saturated fusion system of the group S : Aut F(S).
Finally, the obstruction classes to the existence and uniqueness of centric lin*
*king
systems associated to the saturated fusion system of a group of type S : W , wh*
*ere p -
|W |, live in H*(Z(S); W ) = 0, what proves the corollary under the first assum*
*ption.
If S ~=p1+2+, then p 6= 3, 5, 7, 13, and the result follows from the series o*
*f previous
lemmas.
Therefore it remains to study the cases p = 3, p = 5, p = 7 and p = 13, when
]Fec-rad> 2.
The case p = 3 is worked out in the next lemma.
Lemma 4.12. Let F a saturated fusion system over 31+2+such that ]Fec-rad> 2.
Then F is one in the following list:
______________________________________________
| Out F(31+2+)]Fec-rad AutF (V ) Group |
|____________________________________________|__
| 22 : 2 2 + 2 GL 2(3), GL 2(3)2F4(2)0 |
|_____________________________________________ |
| SD16 4 GL 2(3) J4 |
|____________________________________________|_
where ]Fec-radis the cardinal of Fec-radseparated by F-conjugacy classes and Au*
*tF (V )
is the group of F-automorphisms for each representative of the F-conjugacy clas*
*ses
in ]Fec-rad.
Proof.We have to identify the possible Out F(31+2+) GL 2(3) of order coprime *
*with
3 (Remark 2.3) and containing more than 3 different elements of determinant -1
and trace 0 (Lemma 4.2). There are two subgroups (up to conjugation) verifying
those properties: the group generated by diagonal matrices and 0110, isomorphi*
*c to
22 : 2 ~=D8, and a 2-Sylow of GL 2(3), isomorphic to SD16.
In the first case, Out F(31+2+) ~=22 : 2 divides the 4 rank two elementary ab*
*elian 3-
subgroups in 2 F-conjugacy classes, both of them with 2 elements. We have that *
*all of
the elements AiBj are eigenvectors of eigenvalue -1 in a matrix conjugated to *
*-1001,
so all the rank two elementary abelian p-subgroups can be F-radical. If we look*
* at
14 ALBERT RUIZ AND ANTONIO VIRUEL
the restriction of Aut F(31+2+) to any rank two elementary abelian p-subgroup, *
*there
are matrices of determinant -1, so AutF (V ) ~=GL 2(3). Looking to the informat*
*ion
of the Atlas [4] of the Tits group at p = 3 we deduce that there are two conjug*
*ation
classes of rank two elementary abelian 3-subgroups and both of them are F-radic*
*al.
In the case Out F(31+2+) ~= SD16, the action on the rank two elementary abel*
*ian
3-subgroups is transitive, so there is just one F-conjugacy class. As there is *
*a matrix
conjugated to -1001, then we can make all of them to be radical in our saturat*
*ed
fusion system. Finally, the same argument given in the previous case applies to*
* obtain
that Aut F(V ) ~=GL 2(3). The information in [15, Table (4.1)] tell us that F i*
*s the
fusion system of the Janko's biggest group.
The study of the case p = 5 requires the analysis of one more subgroup of GL*
*2(5).
The analysis is done for any p
Lemma 4.13.
(a) Up to conjugation, there exists just one H GL 2(p) with the following pre*
*senta-
tion:
2-1 p
.
(b) If p 6= 3 and p 6= 7 and Out F(p1+2+) ~=H, then ]Fec-rad= 0
Proof. First we prove (a): The group H has one element y 2 GL 2(p) of order p2-*
*1 so
y will have two different conjugated eigenvalues in Fp2, which will be (p2-1) p*
*rimitive
roots of the unity. All the cyclic subgroups of GL 2(p) of order p2- 1 are conj*
*ugated,
so we fix one of them. Notice that yp+1 = b Id, where b is a primitive (p - 1) *
*root of
the unity in Fp.
We have to prove that fixed y there exists just one possible extension to H.*
* There-
fore we need a matrix x such that x2 = 1 and yxy = bx, an easy calculation tell*
* us
p-1_
that x has determinant -1 and is unique, modulus - Id, but as y 2 = - Id, the
extension H is the same.
In order to prove (b), observe first that we can conjugate H such that it co*
*ntains
the matrix x = 0110(x is an involution with determinant -1, and all of them
are conjugate). Consider y the generator of the cyclic group (p2 - 1) in H, so*
* the
eigenvalues of y are two conjugated primitive roots of unity in Fp2.
Now, if ]Fec-rad> 0 then a subgroup of type (p - 1)-2 must be inside this H.*
* But
the matrices in whose eigenvalues are in Fp are of the form ~ Id, so they d*
*on't
generate any (p - 1)-2. So we have to look to the matrices of the form y0j= yj *
*0110.
One can check that (y0j)2 = bjId, where b is the primitive (p - 1) root of the *
*unity
in Fp that appears in the proof of (a). So if y0jis conjugate to a matrix of th*
*e form
a 0
0 a-2 for a a primitive (p - 1) root of unity in Fp, computing the square we *
*have
a6 = 1 and therefore p = 3 or p = 7.
Now we can study the case p = 5.
p-LOCAL FINITE GROUPS WITH p-SYLOW p1+2+ 15
Lemma 4.14. Let F be a saturated fusion system over 51+2+such that ]Fec-rad> 2.
Then F is one in the following list:
________________________________________
| Out F(51+2+)]Fec-rad AutF (V ) Group |
|______________________________________|__
| 4S4 6 GL 2(5) T h |
|_______________________________________|
where ]Fec-rad is the cardinal of ]Fec-rad separated by F-conjugacy classes and
AutF (V ) is the group of F-automorphisms for each representative of the F-conj*
*ugacy
classes in ]Fec-rad.
Proof.According Lemmas 4.7, 4.13 and 4.10, we have to consider subgroups of GL2*
*(5)
of order prime to 5 such that they project onto A4, S4 or A5 in PGL2(5) . We ge*
*t that
there is just one of those subgroups, isomorphic to 4S4, which projects onto S4.
This group has matrices conjugated to 200-1, so we can consider a saturated *
*fusion
system with radicals, and as all of them are in the same F-conjugation class, a*
*ll 6
will be F-radical.
If we compute the determinant of the restriction of the matrices in Out F(51+*
*2+)
to any rank two elementary abelian p-subgroup, we get that is surjective in F*5*
*, so
AutF (V ) = GL 2(5).
Finally, according to [15, Table (4.1)], F is the fusion system of the Thomps*
*on
group.
The following case, p = 7, must be separated in two different results. The fi*
*rst one,
giving a similar classification as in the previous case, but leaving 3 blank sp*
*aces in
the Group's column. This is because these three saturated fusion systems are ex*
*otic,
that is, there is no finite group with any of these three fusion systems, as it*
* is proved
in the second result, Lemma 4.16.
Lemma 4.15. Let F be a saturated fusion system over 71+2+such that ]Fec-rad> 2.
Then F is one in the following list:
___________________________________________________
| Out F(71+2+)]Fec-rad AutF (V ) Group |
|_________________________________________________|__
| S3 x 3 3 SL2(7) He |
|_________________________________________________|_
| D8 x 3 2 + 2 SL2(7) : 2, SL2(7) :O20N |
|__________________________________________________|
| S3 x 6 3 SL2(7) : 2 He : 2 |
|__________________________________________________|
| S3 x 6 3 + 3 SL2(7) : 2, SL2(7) :F2i0 |
|_____________________________________________24___ |
| 62 : 2 6 SL2(7) : 2 F i24 |
|__________________________________________________ |
| 62 : 2 6 + 2 SL2(7) : 2, GL 2(7) |
|__________________________________________________|
| D16x 3 4 SL2(7) : 2 O0N : 2 |
|__________________________________________________|
| D16x 3 4 + 4 SL2(7) : 2, SL2(7) : 2 |
|__________________________________________________|
| SD32x 3 8 SL2(7) : 2 |
|__________________________________________________|
16 ALBERT RUIZ AND ANTONIO VIRUEL
where ]Fec-radis the cardinal of Fec-radseparated by F-conjugacy classes and Au*
*tF (V )
is the group of F-automorphisms for each representative of the F-conjugacy clas*
*ses
in ]Fec-rad.
Proof. For shortness we do not give details about the computation of the column
Aut F (V ) in each case, but recall that the results are inferred from the meth*
*od de-
scribed in Lemma 4.3.
As Out F(71+2+) GL 2(7) is of order coprime p = 7, we first check the maxi*
*mal
subgroups with this property. From the study of the maximal solvable subgroups
of PGL 2(p) in [10, Proposition 3.3], and Lemma 4.9 we obtain that the maximal
subgroups of GL 2(7) of order coprime with 7 are the preimages of S4 and D2(p 1*
*)by
the projection GL 2(7) ! PGL 2(7), so obtaining the subgroups 6S4, 62 : 2 and 4*
*8 : 2,
which are unique up to conjugation.
Now, we proceed to work with explicit subgroups of GL 2(7), checking in each*
* case
the matrices which can generate subgroups of the form 6V , for V 2 Fec-rad, as
described in Lemma 4.2.
The subgroup 6S4 is then generated by the matrices
6 0 03 36 02 04 32
0 6 , 20 , 34 , 36 , 65 , 11 .
Observe that the determinant of any of those matrices is a square in F7, so any
saturated fusion system F over 71+2+such that OutF (71+2+) S4 verifies ]Fec-r*
*ad= 0.
Now, think of 62 : 2 as the subgroup generated by the diagonal matrices and *
*the
twisting 0110. The subgroups in 62 : 2 of the form 6V are given in the follow*
*ing
table:
_________________________________________________________________________
| V | |V0 V1 V2 V3 V4 V5 V6 V7 |
|_____|_|_______________________________________________________________|_
| | |30 ff 0 3 ff 05 ff 0 1 ff 06 ff 0 2 ff 04 ff 40 ff|
| 6V | |04 3 0 60 2 0 50 1 0 40 03 |
|____|_|________________________________________________________________|_
Then the subgroups of 62 : 2 generated by, at least, three elements in the t*
*able
above are of the form S3 x 3, D8 x 3, S3 x 6 and 62 : 2, and again are unique u*
*p to
conjugation. Therefore, we fix generators for each case and study the specific *
*cases.
The group OutF (71+2+) = S3x3 is generated by 6V1, 6V2and 6V4, and then Fec-*
*rad=
{V1, V2, V3}, all of them in the same F-conjugacy class. The information in [15*
*, Table
(4.1)] tell us that F is the fusion system of the Held group.
The group Out F(71+2+) = D8 x 3 is generated by 6V0, 6V7, 6V1 and 6V6, and t*
*hese
V 's are the only elements in the set Fec-rad, which is divided into two F-conj*
*ugacy
classes of two elements each one. Again the information in [15, Table (4.1)] t*
*ell us
the group with this fusion system: the O'Nan's group.
The group OutF (71+2+) = S3x 6 is generated by 6V1, 6V2, 6V3, 6V4, 6V5 and 6*
*V6. This
is also the list of possible elements in Fec-rad, and is divided into two F-con*
*jugacy
classes of three elements each, so we must consider the possibility of just one*
* of these
conjugacy classes to be F-radical, or both two. In the first case we get the f*
*usion
system of the automorphism group of the Held group, because Out (He) does not
enlarge the cardinal of F-rad, but adds an involution to AutHe(V ). Assume now *
*that
p-LOCAL FINITE GROUPS WITH p-SYLOW p1+2+ 17
the we have six elements in Fec-rad, then by the information in the Atlas [4], *
*F is
the fusion system of the derived Fisher subgroup.
The group OutF (71+2+) = 62 : 2 breaks the rank two elementary abelian 7-subg*
*roups
into two conjugation classes, one with 6 elements, and another with 2. Again t*
*he
information of the Atlas [4] tell that this corresponds to the fusion system of*
* the
Fisher group F i24.
Now we begin the study of the subgroups of 48 : 2 ~=SD32x 3. Observe that we
are in the case of Lemma 4.13, so we consider y an element in GL 2(7) of order *
*48,
and x an involution. We see that the determinant of y must be a generator of F**
*7, and
we can suppose y has determinant 3 (if not, the determinant must be 3-1, and we
can take y-1 as generator). Moreover, we can assume x = 0110. Now we must look
to the matrices in H with eigenvalues 3 and -3: the matrices in with eigenv*
*alues
in F7 are the ones of the form y8i= 3iId, so if there are matrices in 48 : 2 wi*
*th these
properties are of the form yjx. Now using that the determinant must be -2, we g*
*et
that j 2 J def={2, 8, 14, 20, 26, 32, 38, 44}. A direct computation using that *
*xyx = y7,
we get xy = 3y-1x, and so y = -rsst. Then we see that the trace of y2x, which *
*is
equal to the trace of yxy must be 0. Similar arguments show that the trace of y*
*jx
is zero for all j 2 J, so all that matrices have the desired eigenvalues to gen*
*erate a
subgroup of the form 6V . The fact that 7 does not divide the order of 48 : 2, *
*implies
that all that matrices have different eigenvector of eigenvalue 3 (same argumen*
*t as
the proof of Lemma 4.2), so each one generates a different Vi, for all 0 i *
*7.
Now we have to assign to each element yjx a Vij. This will depend on the choi*
*ce of
y, and we can fix one, for example y = 3252. With this we have the following t*
*able:
_________________________________________________________________________
| V | |V0 V1 V2 V3 V4 V5 V6 V7 |
|_____|_|_______________________________________________________________|_
| | |35 ff 0 3 ff 24 ff 4 2 ff 53 ff 3 0 ff 04 ff 40 ff|
| 6V | |04 3 0 35 0 3 42 2 4 40 53 |
|____|_|________________________________________________________________|_
Now we proceed as in the previous case, checking all the subgroups of 48 : 2 ge*
*nerated
by at least three matrices in the list. If we check all the possibilities we ge*
*t D8 x 3
which has been studied previously, D16x 3 and SD32x 3. These last two correspon*
*ds
to the normal subgroup of 48 : 2 generated by and the total group. In t*
*he first
case, checking the action of the outer automorphism group of the O'Nan's group,*
* we
get that the two conjugacy classes in Fec-radfuse in just one, with 4 elements.*
* Finally
SD32x 3 acts transitively on the rank two elementary abelian 7-groups, getting *
*the
result in the list.
Lemma 4.16. There is no finite group with fusion system over 71+2+isomorphic to
any of the following cases:
__________________________________________
| Out F(71+2+)]Fec-rad AutF (V ) |
|__________________________________________|
| D16x 3 4 + 4 SL2(7) : 2, SL2(7) : 2 |
|_________________________________________ |
| 62 : 2 6 + 2 SL2(7) : 2, GL 2(7) |
|_________________________________________ |
| SD32x 3 8 SL2(7) : 2 |
|_________________________________________ |
18 ALBERT RUIZ AND ANTONIO VIRUEL
Proof. According to [2, Lemma 9.2], we have to check finite simple groups and t*
*heir
possible extensions. Checking the Classification of Finite Simple Groups from [*
*6], we
have to look to the fusion systems of the following cases:
(i)Simple groups of Lie type: if the characteristic of this group is 7, then*
*, looking
to the orders, there must be L3(7) or U3(7). The linear one is considered*
* in the
list and the unitary one doesn't have any radical elementary rank two abe*
*lian
7-subgroup.
If the characteristic of the field of the simple group of Lie type is d*
*ifferent
from 7, then we can use the result of [5, 10-2], which says that the 7-Sy*
*low in
that situation has a unique elementary abelian subgroup of maximal rank, *
*and
that is not true for 71+2+.
(ii) Sporadic groups: one can check the list one by one, just in the cases tha*
*t the
7-Sylow is 71+2+and all of them appear in the list of Lemma 4.15.
Finally one has to check the possible extensions, and all the possibilities whi*
*ch give
new information are also in the list.
Lemma 4.17. If p = 13 there is just one saturated fusion system F over p1+2+such
that ]Fec-rad 2, corresponding to the Fischer-Griess monster:
_________________________________________
| Out F(131+2+)]Fec-rad AutF (V ) Group |
|_______________________________________|__
| 3 x 4S4 6 SL2(13.4) M |
|_______________________________________|_
where ]Fec-radis the number of rank two F-radical elementary abelian 13-subgrou*
*ps
in just one conjugation class and AutF (V ) the automorphism group of those.
Proof. To check all the possible saturated fusion systems with more than two F-
radical, using the Lemma 4.10, we get that we need that the projection to PGL 2*
*(13)
must be S4 (it cannot be (D13 1) and we need an element of order 4). The inverse
image of this group, which is unique up to conjugation [10, Proposition 3.3], i*
*s a
subgroup in GL 2(p) isomorphic to 3 x 4S4 and also unique up to conjugation. The
action of this group over the rank two elementary abelian 13-groups divide them
in two conjugacy classes, with 6 and 8 elements and the elements in the first c*
*lass
can be F-radical. The image of the determinant for the matrices which induce an
automorphism in the F-radical rank two elementary abelian 13-group is the ideal
generated by 3 in Z=12, so the automorphisms group for those is SL2(13).4.
To see that it corresponds to the fusion system of M, the Fischer-Griess Mon*
*ster,
we use the information in the Atlas [4]. There is a subgroup 13A in M of order *
*13
such that N(13A) ~=131+2+: (3 x 4S4), so we get the desired Out F(131+2+). Once*
* we
know that, as there is just one conjugation class of such a group in GL 2(13), *
*then it
must break the rank two elementary abelian p-subgroups in 2 orbits, with 6 and 8
elements. So, we have just to prove that there is one F-radical rank two elemen*
*tary
abelian p-subgroup, and it follows from that there is 13B2, a rank two elementa*
*ry
p-subgroup in M, such that N(13B2) ~=132 : 4L2(13).2. The order of 4L2(13).2 is
the same as SL 2(13).4 and there is just one subgroup with this order in GL 2(1*
*3),
p-LOCAL FINITE GROUPS WITH p-SYLOW p1+2+ 19
so it contains SL 2(13). That means that there is at least one F-radical rank *
*two
elementary abelian p-subgroup and the result follows.
Notice that the lemma above corrects a minor error in the description of the *
*13-
fusion of M given in [15, Table (4.1)].
Finally a summary of the arguments followed to prove Theorem 1.1:
Proof of Theorem 1.1.According to Corollary 2.9, it is enough to classify satur*
*ated
fusion systems over p1+2+.
For getting the result for the Table 1, apply Lemmas 4.5, 4.6 and 4.8.
To obtain the data in Table 2, apply Lemmas 4.12, 4.14, 4.15 and 4.17. Lemma
4.16 tell us that the 3 blank spaces left in the "Group" column cannot be fille*
*d.
Finally, to see that there is a complete classification, use that the primes *
*which
doesn't apply the Lemma 4.10 are the ones in the Table 2 and apply again Lemmas
4.12, 4.14, 4.15 and 4.17.
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LAGA, Universit'e Paris XIII, 99 av J.B. Cl'ement, 93430 Villetaneuse, France
E-mail address: ruiz@math.univ-paris13.fr
Dpto de 'Algebra, Geometr'ia y Topolog'ia, Universidad de M'alaga, Apdo corre*
*os
59, 29080 M'alaga, Spain
E-mail address: viruel@agt.cie.uma.es