Classifying finite subgroups of a group

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.

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.

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 $$H$$.