KERNELS OF MAPS BETWEEN CLASSIFYING SPACES
by
Dietrich Notbohm
Abstract. For homomorphisms between groups, one can divide out the kerne*
*l to
get an injection. Here, we develop a notion of kernels for maps between c*
*lassifying
spaces of compact Lie groups. We show that the kernel is a normal subgrou*
*p in a
modified sense and prove a generalization of a theorem of Quillen, namely*
*, a map
f:BG-! BH^pis injective, iff the induced map in mod-p cohomology is finit*
*e. More-
over, for compact connected Lie groups, every map f: BG -! BH^pfactors ov*
*er a
quotient of G in a modified sense and this factorisation is an injection.
1. Introduction.
For a homomorphism ae: G - ! H between groups, we know that the kernel
ker(ae) of ae is a normal subgroup of G, which gives rise to an exact sequence
ker(ae) -! G -! G=ker(ae) :
The induced homomorphism
__ae:G=ker(ae) -! H
is an injection. In this paper we will develop a analoguos concept for maps bet*
*ween
classifying spaces.
To investigate the topological situation we pass to the p-adic completion. *
*We
also allow a more general target. Let G be a compact connected Lie group. A spa*
*ce
X is called BG-local if the evaluation induces an equivalence map(BG; X) ' X
between the mapping space and X. The space X is called almost BG-local if the
evaluation induces an equivalence map(BG; X)const' X between the component
of the constant map const: BG- ! X and X. This is equivalent to the condition
that the loop space X is BG-local.
The situation, we are interested in, is the following
f: BG -! X^pis a map, where G is a compact Lie group, where
(S) X^pis p-complete and almost BZ=p-local and where
H*(X; Fp) is of finite type.
When talking about the kernel of a map f: BG- ! X^p as in (S), one has to
look for elements g 2 G of order a power of p, such that f|B is homotopic to
the constant map. Here, denotes the subgroup generated by g. This leads to
Typeset by AM S-T*
*EX
1
2
a definition of the prekernel, due to Ishiguro [7, 8], and of the kernel, which*
* we
explain now.
For every compact Lie group G there exists a maximal p-toral subgroup SpG,
unique up to conjugation, and every p-toral subgroup is subconjugated to SpG [9*
*].
The group SpG is called the p-toral Sylow group of G. If G is finite, SpG is the
usual p-Sylow group.
For a compact Lie group G, we denote by TG the maximal torus, by NTG the
normalizer of TG , and by WG the Weyl group. Then, SpG is the counterimage of
SpWG in NTG . We also denote SpG by NpTG to indicate that SpG is the p-toral
Sylow subgroup of NTG , too. TG is the component of the unit of NpTG = SpG.
We define a subgroup Sp1 G SpG by the commutative diagram
Tp1 --! Sp1 G --! ss0(SpG)
?? ? ?
y ?y ?y
TG --! SpG --! ss0(SpG) ;
where Tp1 TG denotes the subgroup generated by all elements of order a power
of p. For a map f: BG- ! X^p, we define the prekernel of f as
preker(f) := {g 2 Sp1 G:f|B ' const} ;
and the kernel of f as
ker(f) := cl(preker(f)) ;
where cl( ) denotes the closure in SpG or G.
1.1 Theorem. Let f: BG -! X^pbe a map as in (S).
(1) preker(f) is a subgroup of Sp1 G.
(2) ker(f) is a p-toral subgroup of SpG.
(3) f|Bker(f) is nullhomotopic.
As the proof in the next section shows, the theorem is true without the aasu*
*ption
that H*(X^p; Fp) is of finite type.
These results lead to the following definitions: A map f: BG -! X^p as in (*
*S)
is called injective if ker(f) is the trivial group, and, following [4] or [13],*
* monic
or finite if H*(BG; Fp) is finitely generated over H*(X^p; Fp). In [13] is pro*
*ved
that the kernel of a homomorphism ae: G- ! H between two compact Lie groups is
finite and of order coprime to p if and only if the induced map Bae:BG- ! BH^pis
monic. The following statement also generalizes a result of Dwyer and Wilkerson
[4, Proposition 4.4]
1.2 Theorem. Let f: BG- ! X^p be a map as in (S). Then, f is injective, if and
only if f is monic.
Let O(G) be the orbit category of G, and let Op(G) be the full subcategory,
whose objects are homogenous spaces G=P , where P is a p-toral subgroup of SpG.
Usually, Op(G) is defined to be the full subcategory of the spaces G=P , where P
is any p-toral subgroups of G. Our definition is more convenient for our purpo*
*se
and gives a homotopy equivalent category. This follows because, up to conjugati*
*on,
every p-toral subgroup is contained in SpG [9]. For a subgroup K SpG and a
p-toral subgroup P SpG, we define KP := K \ P .
3
1.3 Lemma. For a p-toral subgroup SpG the following conditions are equiv-
alent:
(1) For every x 2 of order a power of p and for every g 2 G, we have
gxg-1 2 SpG if and only if gxg-1 2 .
(2) For every pair P; P 0 SpG of p-toral subgroups and for every g 2 G, such
that gP g-1 P 0, we have gP g-1 P0.
Proof. Sp1 is a dense subset and contains only elements of order a power
of p. If satisfies the first condition and if gP g-1 P 0, then g(Sp1 \ P )g-1
Sp1 \ P 0. Because P 0is closed, this is also true for , which is condition (*
*2).
Every element of order a power of p generates a finite p-group. Hence, (2) impl*
*ies
(1) obviously.
A p-toral subgroup SpG is called Op(G)-normal, if satisfies one of the
conditions of Lemma 1.3.
1.4 Proposition. Let f: BG- ! X^p be a map as in (S). Then, ker(f) SpG is
Op(G)-normal.
Proof. In [7] is shown that for two subgroups ; 0 SpG, which are conjugated
in G, the restriction f|B is nullhomotopic if and only if f|B0 is nullhomoto*
*pic.
Hence, for every x 2 ker(f), conjugated elements are also contained in ker(f).
For a map f: BG -! X^p as in (S), the kernel ker(f) SpG is not a normal
subgroup of G in general. But, for G connected, ker(f) is the right invariant, *
*which
tells us, on which part the map f: BG- ! X^p is trivial. To avoid discussions a*
*nd
arguments about homotopy colimits, we study Op(G)-normal subgroups of NpTG .
This investigation allows to prove the following theorem:
1.5 Theorem. Let G be a compact connected Lie group and let f: BG- ! X^pbe
a map as in (S). Then, there exists a compact Lie group H and a commutative
diagram
f
BG --! X^p
? fl
q?y flfl
__f
BH^p --! X^p
__
such that f: BH^p-! X^pis injective.
Using a weak form of Theorem 1.1 (1), Ishiguro proved a similiar result for
simple Lie groups. [7, 8]. We remark that we have to take the completion of BH.
Moreover, as the proof shows, the group H in the theorem is not connected in
general. This statement says that, at least, we can divide out by the kernel to
q
make the map injective. Moreover, the homotopy fiber of BG^p-! BH^p is closely
related to ker(f). One might think of this homotopy fiber as being the kernel *
*of
f and of q as a surjection. In general, the homotopy fiber of q might not be t*
*he
completion of the classifying space of a compact Lie group.
4
The paper is organized as follows: In the next section we proof Theorem 1.1,*
* the
third section contains a proof of Theorem 1.2, and in the last section Op(G)-no*
*rmal
subgroups are discussed to prove Theorem 1.5.
Completion are always meant in the sense of [2].
It is pleasure to thank K. Ishiguro for several discussions on this subject.
2. Prekernels and kernels.
We start with the following observation:
2.1 Lemma. Let X^p be a p-complete almost BZ=p-local space. Then, X^p is
almost BG-local for every compact Lie group G.
Proof. A p-complete space X^pis always BZ=p0-local for any prime p0 6= p. If X^p
is also BZ=p-local, then, by [11, x9], follows that X^p is Bss-local for any lo*
*cally
finite group ss. For every compact Lie group G, there exists a mod-p equivalence
Bss- ! BG, where ss is locally finite [6]. So, X^p is BG-local. Because for a*
* p-
complete space X^p, the loop space (X^p) is also p-complete, the same arguments
apply to show that every p-complete almost BZ=p-local space is almost BG-local
for every compact Lie group G.
Let ss be a finite group. For x 2 ss, we define (x) to be the smallest subgr*
*oup of
ss, which is normal in ss and contains x. This is welldefined because the inter*
*section
of two normal subgroups is also normal.
2.2 Lemma. If ss is a finite p-group and noncyclic, then, for every x 2 ss, (x*
*) ss
is a proper subgroup and (x) = .
For a set S of elements of ss, we denote by ~~ the subgroup generated by the
elements of S.
Proof. The center of ss is nontrivial and contains Z=p as subgroup. Hence, the*
*re
q __ __
exists a central extension Z=p!- ss!- ss:= ss=Z=p. If ss ~=Z=pk is cyclic,*
* all
central group extensions are given by abelian groups. Thus , as a noncyclic gro*
*up,
ss ~=Z=p Z=pk and obviously satisfies the statement.
If __ssis noncyclic, we can use an induction over the order of ss. Let __x:*
*= q(x).
Then, q((x)) (__x) 6= __ssby induction hypothesis. This shows that (x) 6= ss.
To prove the second part of the statement, we observe that the group
0(x) := , generated by all the conjugates of x, is normal in ss.
Hence, (x) 0(x). On the other hand, yxy-1 2 (x) for all y 2 ss, which shows
that 0(x) (x).
To prove Theorem 1.1 we need the following result, which may be found in [12*
*],
or [7].
q
2.3 Lemma. Let K!- G!- H be an exact sequence of groups, and let X be an
almost BK-local space. Then,
S
q*: map(BH; X) --! f|BK 'constmap(BG; X)f
is an equivalence.
5
2.4 Proposition. Let f: Bss -! X^pbe a map, where ss is a finite p-group and *
*X^p
a p-complete and almost BZ=p-local space. Let {x1; . .;.xr} be a set of generat*
*ors.
If f|B ' const for all i, then f is homotopically trivial.
Proof. We pove the statement by an induction over the order of ss. If ss ~=Z=pk*
* is
cyclic, there is nothing to show because one of the elements must generate ss. *
*If ss
is noncyclic, by Lemma 2.1, there exists an exact sequence
(x1) -! ss -! __ss:= ss=(x1) :
(x1) is generated by elements of the form yx1y-1 , and f|B ' const.
The order of (x1) is smaller than the order of ss. By induction hypothesis,
f|B(x1) ' const. By Lemma 2.1, X^pSis B(x1)-local, and Lemma 2.2 establishes
an equivalence map(B__ss; X^p) ' g|B(x1)'constmap(Bss; X^p)g. In particuliar*
*, f
__ __
factors over a map f: Bss-! X^p.
The quotient __ssis generated by the elements __xi:= q(xi). The exact sequen*
*ce
(x1) \ -! -! <__xi>
__
and another application of Lemma 2.2 show that f |B<__xi>'_const. Thus we can
again apply the induction hypothesis, which shows that f ' const. This finishes
the proof.
Now, we are prepared to prove Theorem 1.1.
Proof of Theorem 1.1. Let x; y 2 preker(f). We want to show that xy 2
preker(f) or, more generally, that f|B' const.
Sp1 G is a locally finite p group. In particuliar, there exists a sequence
1 2 . . .r . .S.p1G
S
of finite p groups such that Sp1 G = rr [5]. Therefore, is a finite p-*
*group,
and, by the last proposition, f|B' const.SThis proves (1).
Let 0r:= r \ preker(f). Then, preker(f) = r0r. By Proposition 2.3, f|B0r
is nullhomotopic. Because X^p is almost BZ=p-local and hence almost B0r-local
(Lemma 2.1), lim-1ss1(map(B0r; X^p)const ~= lim-1ss1(X^p) vanishes. The Milnor
sequence for calculating the homotopy groups of inverse homotopy limits proves
that f|Bpreker(f)is nullhomotopic.
Lemma 2.5 below shows that Bpreker(f)- ! Bker(f) is a mod-p equivalence.
For the p-complete space X^p, the map [B(ker(f); X^p] -! [B(preker(f); X^p] be-
tween homotopy classes of maps is a bijection [2]. This implies that f|Bker(f)*
* is
nullhomotopic and proves part (3).
ker(f) is the closure of preker(f) in SpG. Thus, the group of the components
of ker(f) is a finite p-group, and ker(f) is a p-toral group which is part (2).*
* This
finishes the proof.
6
2.5 Lemma. Let f: BG- ! X^pbe a map as in (S). Then,the map
B(preker(f)) -! B(ker(f))
is a mod-p equivalence.
Proof. Let T (f) denote the component of the unit of ker(f), T1 (f) the interse*
*ction
of T (f) and Sp1 G, and let ss := ss0(ker(f). These groups fit into the commuta*
*tive
diagram
T1 (f) --! preker(f) --! ss
?? ? fl
y ?y flfl
T (f) --! ker(f) --! ss :
Both rows are exact.
As a locally finite abelian p-group, T1 (f) ~= (Z=p1 )r x A, where A is a fi*
*nite
abelian p-group. Because the closure of T1 (f) is T (f), A is trivial, and T (*
*f) ~=
(S1)r. So, BT1 (f)- ! BT (f) is a mod-p equivalence. The Serre spectral sequence
for mod-p cohomology for the fibrations in the diagram
BT1 (f) --! Bpreker(f) --! Bss
?? ? fl
y ?y flfl
BT (f) --! Bker(f) --! Bss
proves the statement.
3. Injective and monic maps.
In this section we proof Theorem 1.2. Let f: BG- ! X^pbe a map as in (S). Let
Ap(G) denote the Quillen category. The objects are given by elementary abelian *
*p-
subgroups and the morphisms by conjugation in G [13]. To get a finite category,*
* we
take only one object for every isomorphism class of objects, i.e. for every con*
*jugacy
class of a group. The Quillen map
OE:H*(BG; Fp) -! lim- H*(BV ; Fp)
V 2Ap(G)
is H*(BG; Fp)-linear and an F -isomorphism; i.e. kernel and cokernel are nilpot*
*ent
[13, Theorem 7.2]. Let
Y
B := im(OE O f* ) lim- H*(BV ; Fp) H*(BV ; Fp)
V 2Ap(G) V 2Ap(G)
be the image of H*(X^p; Fp).
7
Proof of Theorem 1.2. First, we assume that f is injective and show that f is
monic. For any elementary abelian p-subgroup V G, the restriction f|BV is also
injective. By [4, Proposition 4.4], which is the analogous statement of Theorem
1.2 for elementary abelian p-groups, this implies that H*(BV ; Fp) is a finitel*
*y gen-
eratedQmodule over H*(X^p; Fp) and over B. Because Ap(G) is a finite category,
V 2Ap(G) H*(BV ; Fp) is a finitely generated module over B and a finitely gen*
*er-
ated algebra over Fp. Therefore, B is also a finitely generated algebra over Fp*
* [1,
Proposition 7.8] and hence noetherian. This implies that lim- H*(BV ; Fp), as
Q V 2Ap(G)
a submodule of V 2Ap(G)H*(BV ; Fp), is a finitely generated module over B and
over H*(X^p; Fp).
If f is not monic, i.e the finitely generated Fp-algebra H*(BG; Fp) is not a*
* finitely
generated module over H*(X^p; Fp), there exists an element y 2 H*(BG; Fp) such
that {yi: i 2 N} is a set of linearly independent elements over H*(X; Fp). By
thePabove considerations, for r 2 N big enough, there exists a relation OE(yr) =
r-1 i * ^ r P r-1 i
i=0xiOE(y ) with xi 2 H (Xp ; Fp). That is that y - i=0 xiy is in the ke*
*rnel
of OE and hence nilpotent. Thus, for s 2 N big enough,
r-1X r-1X
0 = (yr - xiyi)s = yrs - x0jyj
i=0 j=0
for suitable x0i2 H*(X^p; Fp). This is a contradiction and proves that f is mon*
*ic.
Now, we assume that f is monic. Let Z=p G be a subgroup of G. Up to
conjugation, Z=p is contained in SpG. By [13], H*(BZ=p; Fp) is finitley generat*
*ed
over H*(BG; Fp), and therefore, also over H*(X^p; Fp). That is that the map
f
BZ=p!- BG!- X^p is homotopically nontrivial. This implies that ker(f) = {1}
and that f is injective.
4. Op(G)-normal subgroups.
For every compact connected Lie group G, there exists a finite covering
ff
K --! eG --! G ;
where Ge ~=GsQx T is a product of a simply connected Lie group Gs and a torus
T . Gs ~= Gi is a product of simply connected simple Lie groups. K is a fin*
*ite
central subgroup of eG. The group eGis unique up to isomorphisms. Such coverings
we call universal finite.
ff
4.1 Lemma. Let K!- Ge!- G be an exact sequence of compact Lie groups, K
finite and eGand G connected. Let f: BG!- X^pbe a map as in (S).
(1) The sequence
SpK!- preker(f O Bff)!- preker(f)
is exact.
(2) ker(f OBff)!- ker(f) is an epimorphism, and ker(f OBff) = Spff-1 ker(f*
*).
8
Proof. Obviously, ff-1 (preker(f)) \ Sp1 eGis contained in preker(f O Bff). In *
*parti-
culiar, SpK preker(f O Bff).SLet := preker(f O Bff)=SpK preker(f). Then,
by Lemma 2.3, map(B; X^p) ' g|BSpK 'constmap(Bpreker(f O Bff); X^p)g. This
implies that f|B is homotopically trivial, and hence, that preker(f), which
establishes the desired sequence of (1).
To prove the second statement we first observe that in this case epimorphism
are maintained under taking closures. The second part in (2) follows from the f*
*acts
that, as a p-toral group, ker(f O Bff) Spff-1 ker(f), and that f|BSpff-1ker(f)*
*is
homotopically trivial.
ff
4.2 Lemma. Let K!- Ge!- G be an exact sequence of compact Lie groups, K
finite and Ge and G connected. Let P SpG be a p-toral subgroup. Then, P is
Op(G)-normal if and only if Spff-1 (P ) SpGe is Op(Ge)-normal.
Proof. Let Qe := ff-1 (P ) and Pe := SpQe. The composition Pe!- Qe!- P is *
*an
epimorphism. This follows, because Pe and Qe have identical components of the
unit, and because, passing to the components, the composition ss0(Pe) = Spss0(Q*
*e)!-
ss0(P ) is an epimorphism of finite groups.
K eGis a central subgroup. The multiplication : (K \ eQ) x eP-! Qe fits into
the pull back diagram
(K \ eQ) x eP --! Qe
?? ?
y ?yff
eP -ff-!P :
Thus, is an epimorphism, and Pe eQ is a normal subgroup. That is that Pe is
the only p-toral Sylow subgroup of eQ.
Every element x 2 P of order a power of p has a lift "x2 eP, also of order a*
* power
of p. Let "g2 eG and g := ff("g). Then, "g"x"g-12 SpGe if and only if gxg-1 2*
* SpG,
and, because eP eQis the only p-toral Sylow subgroup, "g"x"g-12 ePif and only if
gxg-1 2 P . This proves the statement.
Lemma 4.1 and Lemma 4.2 reduce the calculation of kernels and Op(G)-normal
subgroups, G connected, to the case of products of simply connected Lie groups *
*and
tori. Let eG~= Gs x T be such a product. In order to describe Op(Ge)-normal sub-
groups, we associate for every prime p a subgroup H(G; p) to each simply connec*
*ted
simple Lie group G . We define
8
> SU(2) o Z=2 if G = G2 and p = 3
: G else .
Q Q
We define H(Gs; p) := H(Gi; p) for a product Gs = Gi of simply connected
simple Lie groups. Then, BH(Gs; p),! BG is a mod-p equivalence. If p = 3 and
G = G2, this follows from the isomorphism H*(BG2; Z=p) ~=H*(BSU(2); Z=p)Z=2,
and if (p; |WGs |) = 1 from the isomorphism H*(BGs; Z=p) ~=H*(BTGs ; Z=p)WGs .
9
Q
4.3 Proposition. Let eG= Gix T be a product of simply connected simple Lie
groups Gi and a torus T . Let NpTGe be a Op(Ge)-normal subgroup. Then, we
can split Gs = G0x G00such that ~=NpTG0 x ^ and ^ TG00x T . Moreover, ^
is normal in H(G00; p) x T and the image of ^ in G00is finite.
We postpone the proof. This result enables us to prove Theorem 1.5.
Proof of Theorem 1.5. Let f: BG -! X^pbe a map as in (S), and let
K - ! Ge -! G
be a universal finite covering, where Ge ~= Gs x T . By the last proposition a*
*nd
Proposition 1.3, Gs ~=G0x G00and ker(f O Bff) ~=NpTG0 x ^. Now, we define H =
(H(G00; p)xT )=^ . The classical kernel of the projection G0xH(G00; p)xT - ! H *
*is
given by G0x ^, which contains K by construction. We get a commutative diagram
Bi fOBff
B(G0x H(G00; p) x T ) --! BGe ---! X^p
? ? fl
Bf^f?y Bff?y flfl
B_i f
B(G0x H(G00; p) x T )=K --! BG --! X^p
? ? fl
^q?y q?y flfl
__f
BH^p ______BH^p --! X^p :
_
Bi and Bi are mod-p equivalences. BH^pis p-complete, because ss1(BH) is a finite
group [2]. This establishes the map q: BG- ! BH^p. The quotient (G0x ^)=K is a
normal subgroup of (G0x H(G00; p) x T_)=K, and X^pis_almost_B(G0x ^)=K-local
(Lemma 2.1)._Therefore, the map_f O Bi factors_over f:BH^p-! X^p(Lemma 2.3).
Moreover, f O q ' f, because f O ^q' f O Bi. This proves the first half of Theo*
*rem
1.5. fiO^ff
__
By_Lemma_4.1 , ker(f O Bff O Bi) --! ker(f ) is an epimorphism. This shows
that f is injective.
In the rest of this section, we prove Proposition 4.3.
Proof of Proposition 4.3. The subgroup T = \ TGe is invariant under the
Weyl group action. In particuliar, WGe acts on the component e of the unit of ,
and H2(Be; Q) is a WGe-submodule of
M
H2(BTGe; Q) ~= H2(BGi; Q) H2(BT ; Q) :
The first summands are irreducible. Thus, T \ TGi = TGi or the intersection is
trivial.
Let G0 be the product of all factors Gi with TGi , and G00the product of the
other factors of eG. Let x 2 NpTG0 but x 62 TG0. Then x is conjugated to an ele*
*ment
10
in TG0 and therefore, x 2 . This implies that NpTG0 . Moreover, NpTG0 is a
normal subgroup of . In the commutative diagram
i
NpTG0 --! --! ^ := =NpTG0
flfl ? ?
fl ?y ?y
j
NpTG0 --! NpTG0 x NpTG00x T --! NpTG00x T
both rows are central extensions. The map j has a section, which establishes al*
*so a
section for i. Therefore, the upper sequence is the trivial extension, ~=NpTG0*
* x ^,
and ^ NpTG00x T . This is the first part of the statement.
Because e NpTG0 x T , the image of ^ in G00is a finite group and Op(G00)-
normal. We have to investigate finite Op(G)-normal subgroups of simply connected
simple Lie groups. The following proposition finishes the proof.
4.4 Proposition. Let G be a simply connected simple Lie group, and NpTG
a Op(G)-normal finite p-subgroup.
(1) If p divides |WG |, and if G 6= G2 or p 6= 3, then is central in G.
(2) If G = G2 and p = 3, then is central in SU(3) and hence normal in
H(G2; 3).
(3) If (p; |WG |) = 1, then is normal in H(G; p).
Proof. If x 2 then txt-1 x-1 2 for all t 2 TG . Because is finite, all the
commutators are trivial. Thus, x centralizes TG . Because CG (TG ) = TG , = T
TG .
If (p; |WG |) = 1, is normal in NTG = H(G; p). If p divides |WG |, we have
to prove that is central in G or, for G = G2 and p = 3, central in SU(3). Let
x 2 \ Z(G) and xpk = 1. Using this element, we construct an element \ TG ,
which gives a contradiction. This is done by a case by case checking and very m*
*uch
in the flavour of the proof of [7, Theorem 2'].
Before we start, we make two observations. First, if (1) is true for two gr*
*oups
q __
G and H, then it is obviously true for the product G x H and, Second, if G!- _G_
is a covering of connected Lie groups, then (1) is true for G if and only if G
satisfies condition (1). To see this, we consider a finite Op(G)-normal p-subgr*
*oup
NpTG . Then, 0 := <; SpZ(G)> is also a finite Op(G)-normal_p-subgroup.
Because 0 = Sp(q-1 (q(0))), the group q(0)_is Op(G )-normal if and only if 0
is Op(G)-normal (Lemma 4.2), and q(fl0) G is central if and only if 0 G is
central.
Let G = SU(n), n p. Up to conjugation, x can be represented by a di-
agonal matrix D = D(a1; . .;.an), where ai is a pk-th root of unity, and a1 6=
a2. The element y = D(a2; a1; a3; . .;.an) is conjugate to x and thus, xy-1 =
D(a1a-12; a-11a2; 1; . .;.1) 2 . Because a1a-12 6= 1, conjugates of xy-1 gene*
*rate
a subgroup of TG , which contains the maximal elementary abelian p-subgroup VG
of TG . Every element of order p is conjugate to an element in VG . This give*
*s a
contradiction. For G = U(n), n p, the same proof works.
11
Let G = Sp(n), n p. Then, U(n) and SU(2)n are subgroups of maximal rank.
Therefore, TSp(n) is central in U(n) and SU(2)n. But Z(U(n)) \ Z(SU(2)n) =
Z=2 = Z(Sp(n)).
By the above observation, the case of G = Spin(n), n 2p and n 5, can be
reduced to the case of SO(n). Let n = 2k or n = 2k + 1. Then, U(k) SO(2k)
SO(2k + 1) is a subgroup of maximal rank, and is central in U(k). For n 5,
the only WSO(2k) or WSO(2k+1)-invariant subgroup of S1 = Z(U(k)) is Z=2. For
SO(2k), Z=2 U(k) SO(2k) is central, and for SO(2k + 1), Z=2 U(k)
SO(2k + 1) is conjugate to a subgroup in NTSO(2k+1) \ TSO(2k+1), which proves t*
*he
statement for SO(n).
Let G be an exceptional Lie group and p a divisor of |WG |. In this case, we
choose subgroups of maximal rank, given by the following list:
G2 : p = 2; 3 H = SU(3)
F4 : p = 2 H = SU(3) xZ=3 SU(3) p = 3 Spin(9)
E6 : p = 2; 5 H = SU(3)3=Z=3 p = 3 H = SU(2) xZ=2 SU(6)
E7 : p = 3; 5; 7 H = SU(8)=Z=2 p = 2 H = SU(3) xZ=3 SU(6)
E8 : p = 3; 5; 7 H = SU(2) xZ=2 E7 p = 2 H = SU(9)=Z=3 :
Beside the case G = G2 and p = 3, the inclusion H G always induces an
isomorphism SpZ(H) ~= SpZ(G) between the p-Sylow subgroups of the centers.
The data may be obtained from [9], where one can find a complete list of maximal
subgroups of maximal rank of the exeptional Lie groups, and from [14].
Now, we can argue as follows: Let NpTG be an Op(G)-normal subgroup.
Then, by induction over the rank, by the above observations, and by the already
calculated cases, is central in H and hence, central in G.
For G = G2 and p = 3, the argument only shows that is central in SU(3). That
is = Z=3 or is the trivial group. In both cases, is normal in SU(3) o Z=2 =
H(G2; 3). This finishes the last open case and the proof of the proposition.
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