Classifying finite subgroups of a group

From Groupprops
Jump to: navigation, search
This is a survey article related to:subgroups
View other survey articles about subgroups

This article explores the interesting question: given an infinite group, how do we classify all finite subgroups of the group? we'll mainly be looking at linear groups and groups that arise in geometric situations; for instance, fundamental groups or isometry groups of metric spaces and Riemannian manifolds.

General tactics: pass to easier subgroups, quotients and covers

Pass to and from subgroups

Suppose H is a subgroup in a group G. Then we have a natural map from the finite subgroups of G, to the finite subgroups of H, given by intersecting with H:

K \mapsto K \cap H

This map is surjective, because any finite subgroup of H is also a finite subgroup of G. Thus:

  • Going from G to H: To classify the finite subgroups of H, it suffices to first classify the finite subgroups of G, and then check which of them lie inside H.
  • Going from H to G: if we know all the finite subgroups of H, then we can, for each finite subgroup L \le H, try to determine all the possibilities for a finite subgroup K of G such that K \cap H = L. This may not always be easy, but in some cases, it is not hard. For instance, if H is a subgroup of index two in G, then any pre-image of L that is not itself L, is generated by L and exactly one element in G \setminus H. A bit of manipulation can put strong restrictions on what such an element must look like. More generally, the problem is tractable if H is a subgroup of finite index.

Pass to and from quotients by finite normal subgroups

Suppose N is a normal subgroup of G, and H = G/N is the quotient group with the quotient map p:G \to H. We have a natural map from finite subgroups of G to subgroups of H:

K \mapsto p(K)

This map is surjective, because for any finite subgroup L of H, we can take the full inverse image p^{-1}(L). Thus:

  • Going from G to H: To classify the finite subgroups of H, it suffices to classify the finite subgroups of G, and then take the image under p
  • Going from H to G: If we know all the finite subgroups of H, then for each finite subgroup L \le H, try to determine all the possible finite K such that p(K) = L. This is not always easy, but it can be achievable in some cases, particularly the case where N is a finite normal subgroup.

Finding finite-dominating subgroups

What's a finite-dominating subgroup?

A subgroup H of a group G is a finite-dominating subgroup if given any finite subgroup K of G, K is a subconjugate subgroup of H: in other words, there exists g \in G such that gKg^{-1} \le H.

If H is a finite-dominating subgroup in G, then we can reduce the problem of classifying finite subgroups of G, to the problem of classifying finite subgroups of H. Here's how:

  • Classify all finite subgroups of H
  • Then, the finite subgroups of G are simply all the subgroups that can be expressed as conjugate subgroups to the finite subgroups of H.

Finite-dominating collection of subgroups