Classifying finite subgroups of a group
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
This map is surjective, because any finite subgroup of is also a finite subgroup of . Thus:
- Going from to : To classify the finite subgroups of , it suffices to first classify the finite subgroups of , and then check which of them lie inside .
- Going from to : if we know all the finite subgroups of , then we can, for each finite subgroup , try to determine all the possibilities for a finite subgroup of such that . This may not always be easy, but in some cases, it is not hard. For instance, if is a subgroup of index two in , then any pre-image of that is not itself , is generated by and exactly one element in . A bit of manipulation can put strong restrictions on what such an element must look like. More generally, the problem is tractable if is a subgroup of finite index.
Pass to and from quotients by finite normal subgroups
This map is surjective, because for any finite subgroup of , we can take the full inverse image . Thus:
- Going from to : To classify the finite subgroups of , it suffices to classify the finite subgroups of , and then take the image under
- Going from to : If we know all the finite subgroups of , then for each finite subgroup , try to determine all the possible finite such that . This is not always easy, but it can be achievable in some cases, particularly the case where is a finite normal subgroup.
Finding finite-dominating subgroups
What's a finite-dominating subgroup?
If is a finite-dominating subgroup in , then we can reduce the problem of classifying finite subgroups of , to the problem of classifying finite subgroups of . Here's how:
- Classify all finite subgroups of
- Then, the finite subgroups of are simply all the subgroups that can be expressed as conjugate subgroups to the finite subgroups of .