The Z*-theorem for compact Lie groups
Guido Mislin Jacques Thevenaz
Mathematik Institut de Mathematiques
ETH-Zentrum Universite de Lausanne
CH-8092 Z"urich CH-1015 Lausanne
Switzerland Switzerland
Glauberman's classical Z*-theorem is a theorem about involutions of finite *
*groups (i.e.
elements of order 2). It is one of the important ingredients for the classifica*
*tion of finite
simple groups, which in turn allows to prove the corresponding theorem for elem*
*ents of
arbitrary prime order p. Let us recall the statement: if G is a finite group wi*
*th a Sylow
p-subgroup P , and if x is an element of P of order p such that no other G-conj*
*ugate of x
lies in P , then the image of x in G=Op0(G) is central, where Op0(G) denotes th*
*e maximal
normal subgroup of G of order prime to p. The symbol Z*(G) is the classical no*
*tation
for the inverse image in G of the centre of G=Op0(G) and this explains the name*
* of the
theorem.
One can restate the assumption on x in terms of control of fusion. For an a*
*rbitrary
group G and a prime p, we say that a subgroup H of G controls finite p-fusion i*
*n G if the
following two conditions are satisfied:
(a)every finite p-subgroup of G is conjugate to a subgroup of H,
(b)if A is a finite p-subgroup of H and if Ag is also a subgroup of H for some*
* g 2 G,
then g = ch for some h 2 H and c in the centralizer CG (A) of A in G.
This notion is equivalent to the requirement that the inclusion H ! G induces a*
*n equiva-
lence between the categories of finite p-subgroups (in a suitable sense, see Se*
*ction 1). The
assumption on x in the Z*-theorem is then equivalent (at least for finite group*
*s and more
generally for compact Lie groups) to the condition that the centralizer CG (x) *
*controls
finite p-fusion in G (see Proposition 1.8 below). Also the conclusion of the Z**
*-theorem is
readily seen to be equivalent to the equation G = CG (x) . Op0(G) (see Lemma 2.*
*3 below).
- 1 -
It is easy to see (see for instance [Br, Proposition 4]) that one can also *
*restate the
Z*-theorem using the centralizer CG (A) of an arbitrary p-subgroup A. The theor*
*em be-
comes stronger because the centralizer of a larger subgroup is smaller. Explic*
*itely the
statement is now the following.
Z*-THEOREM FOR FINITE GROUPS. Let G be a finite group and p a prime. Let
A be a p-subgroup of G and assume that CG (A) controls (finite) p-fusion in G. *
*Then
G = CG (A) . Op0(G):
The purpose of this paper is to show that the Z*-theorem holds for compact *
*Lie
groups. Recall that a p-toral group is a compact Lie group A whose connected co*
*mponent
A0 is a torus and whose component group A=A0 is a (finite) p-group. Moreover r*
*ecall
that a (not necessarily finite) p-group is a group in which every element has o*
*rder a power
of p. Our main result takes the following form.
Z*-THEOREM FOR COMPACT LIE GROUPS. Let G be a compact Lie group and
p a prime. Let A be either a (not necessarily finite) p-subgroup or a p-toral s*
*ubgroup of G
and assume that CG (A) controls finite p-fusion in G. Then
G = CG (A) . [A; G] = CG (A) . Op0(G):
Moreover [A; G] is a finite normal p0-subgroup of G. In particular the connecte*
*d component
G0 of G centralizes A.
Here [A; G] denotes the subgroup generated by the commutators [a; g] = a-1g*
*-1 ag
for a 2 A, g 2 G. This is always a normal subgroup of G because [a; g]h = [a; h*
*]-1[a; gh].
Note that CG (A) is clearly a closed subgroup of G but Op0(G) need not be c*
*losed
(since for instance it is dense in G when G is the circle group). However Op0(G*
*) is totally
disconnected (since otherwise it would contain a 1-dimensional Lie group), and *
*therefore
its intersection with the connected component G0 of G is necessarily central in*
* G0. Thus
if G0 is a semi-simple Lie group, it has finite centre and consequently Op0(G) *
*is finite in
that case (hence closed).
- 2 -
Our interest in the questions considered in this paper arose from a recent *
*theorem of
the first author [Mi1] giving a cohomological criterion for the control of fini*
*te p-fusion (see
Theorem 1.1 below). However we do not need the full strength of this result her*
*e. In fact
only the easy implication of the theorem is used, namely (a restatement of) the*
* classical
result allowing to compute mod-p cohomology using stable elements.
In view of the fundamental importance of the Z*-theorem for finite groups, *
*we hope
that its generalization can shed some new light on compact Lie group theory, in*
* particular
on the cohomology of these groups. For a first application of the theorem, we *
*refer the
reader to the first author's paper [Mi2].
In the first section of this paper, we give other definitions of control of*
* fusion using
either the category of all p-subgroups or the category of all p-toral subgroups*
*. We prove
that they are all equivalent for compact Lie groups.
For a connected group, or more generally for a group whose component group *
*is a
p-group, we give a direct proof of the Z*-theorem. For the general result howev*
*er, we use
a reduction to the case of finite groups. Thus we need the Z*-theorem for finit*
*e groups,
but we did not succeed in finding a suitable reference for this theorem (althou*
*gh the result
is well known to finite group theorists). It is quoted explicitely (but withou*
*t proof) in
[Pu, Theoreme 1.3]. Of course for p = 2 this is Glauberman's theorem (see for i*
*nstance
[CR, x63C]). For odd p, there is a reduction to the case of simple groups in [B*
*r] and then
the proof essentially consists in a direct inspection of the list given by the *
*classification of
finite simple groups.
As long as no direct proof of the Z*-theorem exists for finite groups and o*
*dd primes,
our result unfortunately depends on the classification of finite simple groups.*
* We regret
to be reduced to adopting this approach, but fortunately our fellow countryman *
*Armand
Borel, facing a similar situation [Bo], has paved the way for a decent excuse b*
*y quoting
B. Shaw:
"You have a low shopkeeping mind. You think of things that would never come*
* into
a gentleman's head."
"That's the Swiss national character, dear lady."
- 3 -
1. Frobenius categories.
Following Puig [Pu], we define for an arbitrary group G and a prime p the F*
*robenius
category Frobp(G) as follows. Its objects are the finite p-subgroups of G and i*
*ts morphisms
are the group homomorphisms induced by conjugation by some element of G. Thus t*
*he
set of morphisms from P to Q is equal to
Mor(P; Q) = CG (P )\TG (P; Q) where TG (P; Q) = {g 2 G | P g Q}:
In particular the set of endomorphisms of P is the group NG (P )=CG (P ) (and e*
*very en-
domorphism is an automorphism). The conjugation by an element g 2 G will be wri*
*tten
Inn(g) : x 7! xg = g-1 xg, and P g= g-1 P g.
Any group homomorphism f : H ! G induces a functor f* : Frobp(H) ! Frobp(G).
When f is the inclusion of a subgroup H in G (which is the only case we conside*
*r in this
paper), then f* is an equivalence of categories if and only if H controls finit*
*e p-fusion
in G. Indeed condition (a) in the definition of the introduction means that any*
* object of
Frobp(G) is isomorphic to an object of Frobp(H), and condition (b) states that *
*f* is full
(while it is clearly always faithful).
For a finite group G, the description of the cohomology H*(G; Z=pZ) in term*
*s of
stable elements in the cohomology of a Sylow p-subgroup allows to prove easily *
*that the
restriction H*(G; Z=pZ) ! H*(H; Z=pZ) is an isomorphism if H controls (finite) *
*p-fusion
in G. For compact Lie groups, the same result holds for the cohomology H*(BG; Z*
*=pZ)
by [FM, Theorem 2.3]. The main result of [Mi1] asserts that the converse also h*
*olds. (The
Frobenius categories are called Quillen categories in [Mi1].) We quote the ful*
*l result for
completeness, although we shall only use the easy part already mentioned for th*
*e proof of
the Z*-theorem.
- 4 -
(1.1) THEOREM. Let f : H ! G be a morphism of compact Lie groups. Then f
induces an equivalence f* : Frobp(H) ! Frobp(G) if and only if the map of class*
*ifying
spaces Bf : BH ! BG induces an isomorphism in mod-p cohomology.
It is convenient to introduce also two other categories of subgroups: the *
*category
Sp(G) of all p-subgroups of G and, in case G is a compact Lie group, the catego*
*ry Tp(G)
of all p-toral subgroups of G. In both cases the morphisms are the group homomo*
*rphisms
induced by conjugation by an element of G. The maximal elements of Sp(G) (viewe*
*d as
a poset) are called Sylow p-subgroup of G (and they always exist by Zorn's lemm*
*a). For
a compact Lie group, the advantage of those two categories compared to Frobp(G)*
* is that
they have maximal elements which are all conjugate (i.e. weak terminal objects).
(1.2) LEMMA. Let G be a compact Lie group.
(a)All Sylow p-subgroups of G are conjugate.
(b)All maximal p-toral subgroups of G are conjugate. The connected component *
*of a
maximal p-toral subgroup is a maximal torus of G.
(c)The closure of a p-subgroup of G is a p-toral subgroup of G.
(d)A Sylow p-subgroup of a p-toral subgroup P is dense in P .
Proof. (b) is proved in [JMO, Lemma A.1]. If A is a p-subgroup of G, then*
* by
[We, 9.4], A contains an abelian normal subgroup B of finite index (because any*
* compact
Lie group is a linear group). Therefore the closure of B is an abelian compact *
*Lie group,
thus a direct product of a torus and a finite abelian group (a p-group in our c*
*ase). It follows
__ __
that the closure A of A is p-toral, proving (c). By (b), we know that A is cont*
*ained in the
normalizer N of a maximal torus. By [Fe, Corollary 1.5], all maximal torsion su*
*bgroups of
N are conjugate. Thus if U and V are two Sylow p-subgroups of G, they are conju*
*gate to
subgroups of Nt where Nt denotes some fixed maximal torsion subgroup of N. But *
*within
Nt all Sylow p-subgroups are conjugate (cf [We, 9.10]). This completes the proo*
*f of (a).
>From (a) and (b) it is clear that every Sylow p-subgroup of a p-toral group P *
*contains the
p-torsion subgroup of the torus P 0; but that subgroup is dense in P 0and (d) f*
*ollows. __|_|
The second fact which will be often used is the following.
- 5 -
*
*__
(1.3) LEMMA. Let A be a p-subgroup of a compact Lie group G and denote by A
the closure of A. Then there exists a countable increasing sequence of finite p*
*-subgroup
S __
Ai of A such that iAi= A. Moreover CG (Ai) = CG (A) = CG (A ) for i sufficien*
*tly large.
__ __
Proof. Since A is p-toral, the torsion elements of (A )0 form a countable *
*subgroup;
thus A is countable. By a result of Schur, since A is a linear torsion group, i*
*t is locally
S
finite (cf [We, 4.9]), and it follows that A = iAi for a suitable increasing *
*sequence of
__ *
* __
finite subgroups {Ai}. Clearly CG (Ai) = CG (A) = CG (A ) for i sufficiently la*
*rge. |_|
The concept of control of fusion extends immediately to our new categories *
*Sp(G)
and Tp(G). We shall say that a subgroup H of G controls p-fusion (respectively *
*controls
p-toral fusion) in G if the inclusion Sp(H) ! Sp(G) (respectively Tp(H) ! Tp(G)*
*) is an
equivalence of categories. The reader can easily rewrite this definition with t*
*wo conditions
(a) and (b) as in the case of the control of finite p-fusion. If C denotes eith*
*er of the three
categories Frobp(G), Sp(G) or Tp(G), we shall also say that H controls fusion i*
*n C to refer
to one of the three types of control of fusion.
(1.4) PROPOSITION. Let G be a compact Lie group and H a closed subgroup of*
* G.
The following conditions are equivalent.
(a) H controls finite p-fusion in G.
(b) H controls p-fusion in G.
(c) H controls p-toral fusion in G.
Proof. Since any finite p-subgroup is a p-toral group, it is clear that (c)*
* implies (a).
To see that (a) implies (b), write a p-subgroup P as a countable union of finit*
*e p-subgroups
S
P = iPi (Lemma 1.3). Since H controls finite p-fusion in G, there exists gi 2*
* G such
that Pigi H. Since G is compact, we can pass to a subsequence and assume that (*
*gi)
converges to some g 2 G. Then any element of P gcan be approximated by an elem*
*ent
of Pigiand since H is closed, it follows that P g H. Now suppose that P and P *
*gare
both subgroups of H, for some g 2 G. Then since H controls finite p-fusion in G*
*, there
exists hi 2 H and ci 2 CG (Pi) such that g = cihi. By Lemma 1.3, there exists *
*i such
that CG (Pi) = CG (P ). Thus g = ch with c 2 CG (P ) and h 2 H. This complete*
*s the
- 6 -
proof of (b). Finally we prove that (b) implies (c) by a continuity argument: *
*if Q is a
p-toral subgroup of G, choose a dense p-subgroup P in Q. Since H controls p-fus*
*ion in G,
there exists g 2 G such that P g H. Then Qg H because H is closed. Similarly*
* if
both Q and Qg are subgroups of H, for some g 2 G, then since H controls p-fusio*
*n in G,
there exists h 2 H and c 2 CG (P ) such that g = ch . But CG (P ) = CG (Q) and*
* thus
c 2 CG (Q). __|_|
For any category C of subgroups (such as Frobp(G), Sp(G) or Tp(G)), we shal*
*l say
that a subgroup A 2 C is isolated in C if for each object P 2 C, there is at mo*
*st one
morphism from A to P . Translating this condition, we see that A is isolated in*
* C if and
only if for every h; g 2 G such that < Ah; Ag > is contained in a subgroup in C*
*, the
element gh-1 centralizes A. Here < Ah; Ag > denotes the subgroup generated by *
*Ah
and Ag. Conjugating by h-1 and replacing gh-1 by g, we see that actually A is i*
*solated
in C if and only if whenever < A; Ag > is contained in a subgroup in C, then g *
*2 CG (A).
When C = Frobp(G), if an isolated subgroup A is generated by a single element x*
*, finite
group theorists often say that x is weakly closed in a Sylow p-subgroup. This *
*condition
corresponds to the assumption of the classical statement of the Z*-theorem. Any*
* central
subgroup belonging to C is isolated in C. Also if an isolated subgroup A is con*
*tained in a
group P 2 C, then the definition immediately implies that A is central in P .
It is obvious that if A 2 Frobp(G) is isolated in Sp(G), then A is isolated*
* in Frobp(G).
We now show that for a compact Lie group, the converse holds.
(1.5) LEMMA. Let G be a compact Lie group and let A 2 Frobp(G). Then A is
isolated in Frobp(G) if and only if A is isolated in Sp(G).
Proof. Assume A is isolated in Frobp(G). Let g 2 G be such that P =< A; Ag *
*> is a
p-group. In order to prove that g centralizes A, it suffices to show that P is *
*finite and then
apply the assumption. As observed earlier, P is locally finite since it is a l*
*inear torsion
group [We, 4.9]. But P is finitely generated, hence finite. __|_|
Now we come to the link between the definition of isolated subgroups and co*
*ntrol of
fusion.
- 7 -
(1.6) LEMMA. Let G be an arbitrary group, let C be any of Frobp(G), Sp(G) *
*or
Tp(G) (with G a compact Lie group for the latter case) and let A 2 C. If CG (A)*
* controls
fusion in C, then A is isolated in C.
Proof. The argument in the three cases is the same. Suppose < A; Ag > P whe*
*re
P 2 C. By control of fusion, there exists x 2 G such that P x CG (A). For a 2 A*
*, we
have ax; agx 2 CG (A) CG (a) and also a 2 CG (a). Clearly CG (a) also control*
*s fusion
(because CG (A) CG (a)) and applying this to the morphism Inn(x) : < a > ! < a*
*x >,
we obtain x 2 CG (a). Similarly gx 2 CG (a) and therefore g 2 CG (a). This hold*
*s for all
a 2 A, showing that g 2 CG (A). Thus A is isolated in C. __|_|
When all maximal elements of our category are conjugate, the converse of Le*
*mma 1.6
holds. We only give the argument for compact Lie groups.
(1.7) LEMMA. Let G be a compact Lie group.
(a)Let A 2 Sp(G). If A is isolated in Sp(G), then CG (A) controls p-fusion.
(b)Let A 2 Tp(G). If A is isolated in Tp(G), then CG (A) controls p-toral fusi*
*on.
Proof. Let C be either Sp(G) or Tp(G) and let P 2 C. Since all maximal elem*
*ents of C
are conjugate (Lemma 1.2), there exists g 2 G such that A and P glie in such a *
*maximal
element Q. Since A is isolated, it follows that Q centralizes A. Therefore P g *
*CG (A),
proving the first condition for control of fusion.
Now suppose that P; P g CG (A) for some g 2 G. Thus we have A; Ag-1 CG (P *
*).
But CG (P ) is a compact Lie group, so all its Sylow p-subgroups (respectively *
*maximal
p-toral subgroups) are conjugate. Therefore there exists c 2 CG (P ) such that*
* A and
Ag-1c lie in such a maximal element. Since A is isolated, g-1 c centralizes A. *
*Therefore
g 2 CG (P ) . CG (A), proving the second condition for control of fusion. __|_|
Collecting the results above, we obtain the following proposition.
- 8 -
(1.8) PROPOSITION. Let G be a compact Lie group and let A be either a p-su*
*bgroup
or a p-toral subgroup of G. Then the following conditions are equivalent.
(a)CG (A) controls finite p-fusion.
(b)CG (A) controls p-fusion.
(c)CG (A) controls p-toral fusion.
If A is finite, then these conditions are also equivalent to the following *
*ones.
(d)A is isolated in Frobp(G).
(e)A is isolated in Sp(G).
(f)A is isolated in Tp(G).
Recall that the second set of conditions can always be realized since by Le*
*mma 1.3,
one can replace A by a finite subgroup without changing its centralizer.
The following corollary will be crucial in the proof of the Z*-theorem.
(1.9) COROLLARY. Let H be a closed subgroup of a compact Lie group G and l*
*et
A H be a p-subgroup or a p-toral subgroup. If CG (A) controls finite p-fusion *
*in G, then
CH (A) controls finite p-fusion in H.
Proof. We first replace A by a finite p-subgroup B of A such that CG (A) = *
*CG (B) and
CH (A) = CH (B). If CG (B) controls finite p-fusion in G, then B is isolated in*
* Frobp(G)
and therefore B is also isolated in the subcategory Frobp(H). But by Propositio*
*n 1.8, this
implies that CH (B) controls finite p-fusion in H. __|_|
- 9 -
2. Proof of the Z*-theorem.
We first treat the following special case.
(2.1) PROPOSITION. Let G be a compact Lie group and assume that G=G0 is a
p-group. If A is a p-group or a p-toral subgroup of G such that CG (A) control*
*s finite
p-fusion in G, then CG (A) = G, that is, A is central in G.
Proof. By Lemma 1.3 we can assume that A is a finite p-group. Write K =
CG (A). Since G=G0 is a p-group, K=K0 is a (finite) p-group too (cf. [Ad, Lem*
*ma 7.1]
or [JMO, Proposition A.4]). By Theorem 1.1, the induced map BK ! BG is a mod-p
cohomology isomorphism. First we claim that K covers the quotient G=G0. Other*
*wise
K . G0 is contained in a maximal subgroup M of G of index p. Let ff : G=G0 ! Z*
*=p
be a homomorphism with kernel M=G0. Then ff 2 H1(G=G0; Z=p) is non-trivial but *
*its
restriction to K=K0 is trivial. Inflating this to the cohomology of BG and BK (*
*inflation
is injective for H1), we obtain a non-trivial element in the kernel of the rest*
*riction from
BG to BK, against our assumption. Thus K=K0 ~=G=G0 as claimed.
Now we only have to prove that K0 = G0. Note that the fibration G ! EG ! BG
implies that the group ss = ss1(BG) is isomorphic to ss0(G) = G=G0, and this is*
* a p-group
by assumption. Moreover by the first part of the proof, the inclusion K ! G ind*
*uces an iso-
morphism ss1(BK) ~=ss1(BG) = ss. If M is any finitely generated Z=p[ss]-module,*
* then M
has a finite filtration by submodules Mi (with 1 i n) such that the quotients*
* Mi=Mi+1
are trivial Z=p[ss]-modules; indeed the trivial module Z=p is the only simple Z*
*=p[ss]-module
since ss is a p-group. Now we claim that the map H*(BG; M) ! H*(BK; M) of co-
homology with local coefficients is an isomorphism; indeed since BK ! BG induce*
*s an
isomorphism in cohomology with trivial coefficients by Theorem 1.1, the claim f*
*ollows by
induction on the length of the filtration, using the long exact sequence of coh*
*omology as-
sociated to the sequence 0 ! M1 ! M ! M=M1 ! 0. Now we wish to apply this to the
free module Z=p[ss] = Indss1(Z=p). By Shapiro's lemma,
H*(BG; Z=p[ss]) ~=H*(BG0; Z=p) ; and similarly H*(BK; Z=p[ss]) ~=H*(BK0; Z=*
*p) :
- 10 -
It follows that the map H*(BG0; Z=p) ! H*(BK0; Z=p) is an isomorphism. But sin*
*ce
G0 ' (BG0) and K0 ' (BK0) and since the spaces BG0 and BK0 are simply
connected, the map H*(G0; Z=p) ! H*(K0; Z=p) is an isomorphism too. Now K0 and *
*G0
are compact orientable manifolds, so one can conclude that they have the same d*
*imension.
It follows that K0 = G0 since they are both connected. Therefore K = G and the *
*proof is
complete. __|_|
For the proof of the general case, we will also need the following lemma. A*
*lthough it
is certainly well known, we provide a proof for the convenience of the reader.
(2.2) LEMMA. Let G be a compact Lie group and A a finite p-subgroup of G. *
*Then
there exists a finite subgroup F containing A which maps onto G=G0.
Proof. As observed in Lemma 1.2, A normalizes a maximal torus T of G. It is*
* well
known that the normalizer N = NG (T ) maps onto G=G0 and that any maximal torsi*
*on
subgroup of N is dense in N. Hence we can choose a torsion subgroup Ntof N cont*
*aining A
and mapping onto G=G0. Since Nt is locally finite (being a linear torsion group*
* [We, 4.9]),
and since A and G=G0 are finite, we can find a finite subgroup F of Nt containi*
*ng A and
mapping onto G=G0. __|_|
We also need the following result which was partially mentioned in the intr*
*oduction.
It shows the equivalence between several forms of the conclusion of the Z*-theo*
*rem.
(2.3) LEMMA. Let A be a p-subgroup of a finite group G. The following cond*
*itions
are equivalent.
(a)The image of A in G=Op0(G) is central.
(b)G = CG (A) . Op0(G).
(c)G = CG (A) . [A; G] and [A; G] is a p0-group.
Proof. It is obvious that (c) implies (b) and that (b) implies (a). Assum*
*e now (a).
Then clearly [A; G] Op0(G) so that N = [A; G] is a p0-group. Let ss : G ! G=N.*
* We
first show that G = NG (A) . N. Let g 2 G. Since ss(g) centralizes ss(A), we ha*
*ve
Ag ss-1 (ss(A)) = N . A :
- 11 -
Since both A and Ag are Sylow p-subgroups of N . A, we have Ag = An for some n *
*2 N
and therefore gn-1 2 NG (A). Now we show that NG (A) = CG (A). If h 2 NG (A) th*
*en for
each a 2 A, the commutator [a; h] belongs to A. But this commutator also belong*
*s to N
and since A \ N = 1, it follows that h centralizes a. __|_|
Proof of the Z*-theorem. By Lemma 1.3, we can choose a finite p-subgroup B *
*of A
such that CG (B) = CG (A). The image of B in G=G0 is a p-group, whose inverse i*
*mage
in G is a compact Lie group K with component group a p-group. Since CG (B) cont*
*rols
finite p-fusion in G by assumption, CK (B) controls finite p-fusion in K by Cor*
*ollary 1.9,
and by Proposition 2.1 we obtain that B is central in K, and in particular G0 c*
*entralizes B.
Choose now a finite subgroup F mapping onto G=G0 and containing B (Lemma 2.*
*2).
By Corollary 1.9 again, CF (B) controls (finite) p-fusion in F . By the Z*-the*
*orem for
finite groups, it follows that we have F = CF (B) . Op0(F ) and therefore by Le*
*mma 2.3,
F = CF (B).[B; F ] and [B; F ] is a p0-group. But since G = G0.F = G0.CF (B).[B*
*; F ] and
since G0 centralizes B, we conclude that G = CG (B) . [B; F ]. Note that [B; F *
*] = [B; G]
because G = G0 . F and G0 centralizes B. Thus G = CG (B) . [B; G], and a forti*
*ori
G = CG (A) . [A; G] and G = CG (A) . Op0(G) since [B; G] is contained in both [*
*A; G] and
Op0(G). This completes the proof of the main statement of the Z*-theorem.
__
It remains to show that [A; G] is a finite p0-group. Let A be the closure o*
*f A. Since
__ __ __ *
* __
CG (A) = CG (A ) we have G = CG (A ) . Op0(G) and so [A ; G] lies in Op0(G). Th*
*us [A ; G]
__0 __ *
* __0
is totally disconnected and therefore [A ; G] and [A ; G0] are trivial groups *
*(because [A ; g]
__
and [a; G0] are connected, hence trivial, for all g 2 G and a 2 A). If U and V *
*are finite
__ __0
subgroups such that A = A .U and G = G0.V , then by using standard rules for ex*
*panding
__ __
commutators we obtain [A ; G] = [U; V ]. It follows that [A ; G] is finitely ge*
*nerated. Since it
is also a subgroup of the torsion group Op0(G), we apply once again Schur's res*
*ult [We, 4.9]
__ __
to deduce that [A ; G] and its subgroup [A; G] are finite. |_|
For a compact Lie group G, let Z*(G) be the inverse image in G of Z(G=Op0(G*
*)).
Since there is a unique maximal torsion subgroup in Z*(G) (the inverse image of*
* the torsion
subgroup of the abelian group Z(G=Op0(G)) ) and since all Sylow p-subgroups of *
*this
torsion subgroup are conjugate [We, 9.10], all Sylow p-subgroups of Z*(G) are c*
*onjugate.
- 12 -
We denote by Z*(G)p an arbitrary Sylow p-subgroup of Z*(G). It is not difficult*
* to show
(using arguments similar to those of Lemma 2.3) that any subgroup of Z*(G)p is *
*an isolated
p-subgroup of G (i.e. isolated in Sp(G)). Since conversely the Z*-theorem asser*
*ts that any
isolated p-subgroup of G is contained in Z*(G), we see that Z*(G)p and its conj*
*ugates are
precisely the maximal isolated p-subgroups of G.
Denote by Gp (respectively Gp-tor) a Sylow p-subgroup of G (respectively a *
*maximal
p-toral subgroup of G). We now combine Theorem 1.1 (this time using its full st*
*rength)
with the results of the present paper.
(2.4) COROLLARY. Suppose that f : H ! G is a morphism of compact Lie groups
inducing a mod-p cohomology isomorphism. Then f induces isomorphisms Hp ~= Gp ,
Hp-tor~=Gp-torand Z*(H)p ~=Z*(G)p.
Proof. By Theorem 1.1, f induces an equivalence of categories Frobp(H) ! Fr*
*obp(G).
This easily implies that the restriction of f to Hp is injective and that f(Hp)*
* is a Sylow
p-subgroup Gp of G; therefore Hp ~=Gp . Taking the closure of Hp and Gp , we o*
*btain
Hp-tor~= Gp-tor. Now the inclusion f(H) ! G also induces a mod-p cohomology is*
*o-
morphism, so by Theorem 1.1, f(H) controls finite p-fusion in G. By Propositio*
*n 1.4,
f(H) also controls p-fusion in G and the equivalence Sp(H) ! Sp(G) has to map m*
*aximal
isolated objects to maximal isolated objects; this implies that Z*(H)p ~=Z*(G)p*
*. __|_|
- 13 -
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