Posted this to MSE several weeks ago and it got 3 upvotes but no answers or even comments so I'm cross-posting to MO

Aschbacher's theorem says that every maximal subgroup of a finite simple classical group falls into at least one of the 9 Aschbacher classes.

Is there a similar result for compact simple classical groups?

That is, do all maximal finite subgroups of compact simple classical groups fall into one of the 9 Aschbacher classes?

Motivation: This paper https://arxiv.org/pdf/math/0502080.pdf seems to use some sort of Aschbacher's theorem for complex matrix groups in the discussion after prop 1.2 (on page 2).

Is this one of those situations where representation theory of finite groups over a field of characteristic not dividing $ |G| $ is equivalent to representation theory over a field of characteristic $ 0 $? So the representation theory for finite groups actually encompasses the representation theory over $ \mathbb{R} $ and $ \mathbb{C} $?


1 Answer 1


Every representation of a finite group in characteristic $0$ is equivalent to one over a finite extension of ${\mathbb Q}$ (i.e. a number field), so I guess if your original field for the compact group is ${\mathbb R}$ or ${\mathbb C}$ then the representation is equivalent to one over a proper subfield, which is one of the Aschbacher categories. But there is a difference from the finite field case in that the Schur index can be nontrivial in characteristic $0$, which means that, unlike in the finite field case, the minimal field over which it can be represented is not always unique.

But, as you said yourself, Aschbacher's theorem is an attempt to classify maximal subgroups of (almost simple extensions of) classical groups over finite fields, but your question seems to be referring to all finite subgroups, not just the maximal ones. The definitions of the geometric type classes $\mathcal{C}_i$ in Aschbacher's paper include extra conditions that are intended to exclude subgroups that are definitely not maximal. For example $\mathcal{C}_2$ does not contain all imprimitive groups: there are extra conditions, which depend on the classical group under consideration. The main theorem is that any maximal subgroup of a classical group lies in (at least) one of these classes.

In their book "The Subgroup Structure of the Finite Classical Groups", Kleidman and Lieback prove a much more precise version of the theorem in dimensions greater than $12$, and they made small adjustments to the definition some of the $\mathcal{C}_i$, to exclude other types of subgroups that turned out not to be maximal. The corresponding result for dimensions up to $12$ was proved later by Bray, Holt and Roney-Dougal.

I think the result that you are asking about in your question is a much weaker version of Aschbacher's theorem, in which the classes are simply reducible groups, imprimitive groups, etc, without any extra conditions. Various people have pointed out that this weaker version should not be attributed entirely to Aschbacher, because similar results were known previously. In his book "The Finite Simple Groups", Robert Wilson refers to this result as the Aschbacher-Dymkin theorem, because there is a 1952 paper in Russian by Dynkin, which apparently proves something similar.

We can assume that the representation is absolutely irreducible, since otherwise we are in the first or third of the Aschbacher classes (in the weaker version). Then the main idea of the proof is to consider a minimal non-scalar normal subgroup $N$ of the group $G$, and apply Clifford's Theorem. Then if the decomposition of the restriction of the representation to $N$ is not homogeneous, the group is imprimitive, and if it is homogeneous but not irreducible, we get a decomposition as a tensor product.

If the restriction to $N$ is irreducible and $NZ(G)/Z(G)$ is elementary abelian, then $N$ is extraspecial or of symplectic type, and we are in the sixth Aschbacher class. If it is a direct product of more than one nonabelian simple group, then we are in the seventh class (symmetric tensor decomposition). Finally, if $NZ(G)/Z(G)$ is simple, then $G$ is nearly simple, and we are in the eighth of ninth class.

I think all or at least most of those arguments apply also in characteristic $0$. But there are some subtle differences. For example, a reducible but not absolutely irreducible real representation can decompose as two or more isomorphic representations over $\mathbb{C}$, which does not happen in the finite field case, again because the Schur index is always $1$ over finite fields.

  • $\begingroup$ I'll definitely read over this way more thoroughly later but wow looks like a very juicy and answer and yes you are right I meant all maximal finite subgroups not all finite subgroup $\endgroup$ Apr 7 at 2:38
  • $\begingroup$ I think it is fair to say that a lot of the Clifford theoretic reductions you outline can be said to originate in the study of finite subgroups of GL(n,C). $\endgroup$ Apr 10 at 7:57
  • $\begingroup$ The basic (fairly classical) question there being " which finite (complex) linear groups can not be described in terms of lower dimensional linear groups?" $\endgroup$ Apr 10 at 8:29
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    $\begingroup$ The answer being (roughly) those G with F*(G) = Z(G)L, where L is either quasisimple or extra special with L/Z(L) minimal normal in G/Z(G) . $\endgroup$ Apr 10 at 8:35

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