p-STUBBORN SUBGROUPS OF
CLASSICAL COMPACT LIE GROUPS
by Bob Oliver
For any compact Lie group G, let O(G) denote the "orbit category" of G: the
category whose objects are the orbits G=Hfor closed subgroups H G,and whose
morphisms are all G-maps between orbits. For any prime p, let Rp(G) denote
the category of those orbits G=P for p-stubborn subgroups P G: namely, those
subgroups which satisfy the conditions
(a) P is p-toral (i.e., an extension of a torus by a finite p-group)
(b) N(P)=P is finite
(c) N(P)=P is p-reduced: there is no nontrivial normal p-subgroup 1 6= Q C
N(P )=P .
One of themain results in [JMO] is a decomposition for BG indexed over Rp(G).
More precisely, for any G and p, the natural projection map
hocolim!(EG=P ) !BG
G=P 2Rp(G)
induces an equivalence of Fp-homology [JMO, xx1-2]. This decomposition of BG,
and the category Rp(G), play a central role in [JMO] and [JMO2] as a tool for
describing sets of homotopy classesof maps from BG to BH for any (other) compact
connected Lie group H. But in most cases, explicit descriptions of the Rp(G),or
explicit lists of the p-stubborn subgroups ofG, were not necessary to obtain the
results in those papers. However,recent results of Notbohm [N], proving in many
cases the uniqueness of the completed classifying spaces BG^p(uniqueness among
spaces with the same mod p cohomology),do require a more precise description of
the p-stubborn subgroups of the classical compact Lie groups. And this provides
the motivation for the present paper.
In Theorems 3 below, the p-stubborn subgroups of the matrix groups U (n),O (*
*n),
and Sp(n) are described explicitly for each n and p. In Theorems 5 and 7, these
results are extended to describe the p-stubborn subgroups of SU (n) and SO (n).
In all cases, the p-stubborn subgroups show a surprisingly simple pattern: being
generated from a small collection of "basic" p-stubborn subgroups by products a*
*nd
wreath products.
Typeset by AMS-TEX
For a finite group G, it turns out that the "p-stubborn" subgroups are preci*
*sely
the same as the "radical" p-subgroups which are used by group theorists when
classifying finite simple groups. The description given here of p-stubborn subg*
*roups
of the classical matrix groups is very similar in nature to the description of *
*radical
subgroups of symmetric groups found by Alperin & Fong in [AF, Theorem 2A].
The following fundamental properties of p-stubborn subgroups will be used fr*
*e-
quently.
Lemma 1. Let P be a p-stubborn subgroup of a compact Lie group G.Then the
following two properties hold.
(i) Any p-toral subgroup H G which is normalized by N(P) (i. e., N(P) N(H))
is contained in P.
(ii) CG0(P ) Z(P ), and CG(P ) = Z(P) if G=G0 is a p-group.
Proof. Part (ii) is shown in [JMO, Lemma 1.5(ii)]. Part (i) is essentially show*
*n in
the same lemma, but in a slightly different formulation. For that reason, we re*
*peat
the proof here.
Assume that H * P ,and that N (P ) normalizes H .Set H 0= hH;P i % P . Since
P normalizes H, H is normal in H0, and H0is p-toral since H0=H is a quotient
group of P . Also, N(P ) N(H0); and
Ker[N(P )=P! N (H0)=H 0] = (H 0"N(P ))=P = NH0(P )=P
is a nontrivial normal p-subgroup ofN (P)=P (cf. [JMO, Lemmas A.2 & A.3]).
Which contradicts the assumption that P is p-stubborn.
We now define certain p-stubborn subgroups of n, O(n), U (n) and Sp(n): sub-
groups which will be seen to generate all other p-stubborn subgroups of such gr*
*oups.
In the following definitions, oe0; : :;:oek1 2pk will denote the permutatio*
*ns
aei + pr if i j 1; : :;:(p 1)pr (mo dpr+1 )
oer(i) =
i (p 1)pr if i j (p 1)pr+ 1; : :;:pr+1 (mod pr+1):
Note that these generate an elementary abelian p-subgroup of rank k. Also,
i = e2ssi=p denotes a primitive p-th root of unity. Define matrices
A1; : :;:Ak1 ;B1; : :;:Bk1 2U (pk) by setting
ae i[(i1)=pr] if i =j ae1 ifoe (i) = j
(Ar)ij= and (Br)ij= r
0 if i 6=j 0 ifoer(i) 6= j
(where [] denotes greatest integer).The Ar are thus all diagonal matrices, and *
*the
Br are thep ermutation matrices for the oer. Note that they satisfy the commuta*
*tor
relations
[Ar; As] = I= [Br; Bs] = [Br;As] (r 6= s); and [Br; Ar] = i I: