Classifying finite subgroups of a group

From Groupprops

This is a survey article related to:subgroups
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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 $H$ .