Conjugate-intersection index theorem
This article describes an easy-to-prove fact about basic notions in group theory, that is not very well-known or important in itself
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Let be a subgroup of of finite index in . Let be distinct conjugates of in , so that . Let . Then:
- is of finite index in
- The index of in is bounded from above by viz. or .
- Poincare's theorem bounds the index of the normal core of a subgroup, in terms of the index of the subgroup
We have a natural action of on the coset space by left multiplication. Now, consider the set of -tuples over with all entries distinct. acts on this set by the coordinate-wise action on .
Now, for each conjugate , find a such that .
Claim: The group is the isotropy subgroup for the element
Proof: Note that the isotropy subgroup for this point under the coordinate-wise action must stabilize every coordinate, and conversely, if a group element fixes every coordinate, it must belong to the isotropy subgroup. Hence, the isotropy subgroup is the intersection of the isotropy subgroups for every coordinate. But the isotropy subgroup for is precisely . This gives the result.