local finiteness is closed under extension, proof that
Let be a group and a normal subgroup of
such that and are both locally finite
.
We aim to show that is locally finite.
Let be a finite subset of .
It suffices to show that is contained in a finite subgroup of .
Let be a set of coset representatives of in , chosen so that . Let be the function mapping cosets to their representatives, and let be defined by for all . Let be the canonical projection. Note that for any we have .
Put , which is finite as is locally finite. Let , let and let
Put , which is finite as is locally finite. Note that and .
For any we have .
Note that ,
and so every element of is a product of elements of .
So any element of the form , where and ,
is a product of elements of the form
for and ;
but for some and ,
so is a product of elements of the form
,
and therefore .
We claim that .
Let and .
We have .
But, by the previous paragraph, and ,
so .
Thus .
Also, .
But ,
so .
Thus .
It follows that is a subgroup of , and it is clearly finite.
For any we have .
So , which completes the proof.
Title | local finiteness is closed under extension |
---|---|
Canonical name | LocalFinitenessIsClosedUnderExtensionProofThat |
Date of creation | 2013-03-22 15:36:53 |
Last modified on | 2013-03-22 15:36:53 |
Owner | yark (2760) |
Last modified by | yark (2760) |
Numerical id | 6 |
Author | yark (2760) |
Entry type | Proof |
Classification | msc 20F50 |
Related topic | LocallyFiniteGroup |