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.
Contents
General tactics: pass to easier subgroups, quotients and covers
Pass to and from subgroups
Suppose is a subgroup in a group
. Then we have a natural map from the finite subgroups of
, to the finite subgroups of
, given by intersecting with
:
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
Suppose is a normal subgroup of
, and
is the quotient group with the quotient map
. We have a natural map from finite subgroups of
to subgroups of
:
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?
A subgroup of a group
is a finite-dominating subgroup if given any finite subgroup
of
,
is a subconjugate subgroup of
: in other words, there exists
such that
.
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
.