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, proof that|
|Date of creation||2013-03-22 15:36:53|
|Last modified on||2013-03-22 15:36:53|
|Last modified by||yark (2760)|