local finiteness is closed under extension, proof that

Let G be a group and N a normal subgroupMathworldPlanetmath of G such that N and G/N are both locally finitePlanetmathPlanetmathPlanetmath. We aim to show that G is locally finite. Let F be a finite subset of G. It suffices to show that F is contained in a finite subgroup of G.

Let R be a set of coset representatives of N in G, chosen so that 1R. Let r:G/NR be the function mapping cosets to their representatives, and let s:GN be defined by s(x)=r(xN)-1x for all xG. Let π:GG/N be the canonical projection. Note that for any xG we have x=r(xN)s(x).

Put A=r(π(F)), which is finite as G/N is locally finite. Let B=s(FAAA-1), let C=BB-1 and let

D={a-1caaA and cC}N.

Put H=D, which is finite as N is locally finite. Note that 1AR and 1BCDHN.

For any a1,a2A we have a1a2=r(a1a2N)s(a1a2)AB. Note that D-1=D, and so every element of H is a productPlanetmathPlanetmathPlanetmath of elements of D. So any element of the form a-1ha, where aA and hH, is a product of elements of the form a-1a1-1ca1a for a1A and cC; but a1a=a2b for some a2A and bB, so a-1ha is a product of elements of the form b-1a2-1ca2b=b-1(a2-1ca2)bCDBH, and therefore a-1haH.

We claim that AHG. Let a1,a2A and h1,h2H. We have (a1h1)(a2h2)=a1a2(a2-1h1a2)h2. But, by the previous paragraph, a1a2AB and a2-1h1a2H, so a1a2(a2-1h1a2)h2ABHHAH. Thus AHAHAH. Also, (a1h1)-1=h1-1a1-1Ha1-1. But a1-1=r(a1-1N)s(a1-1)AB, so Ha1-1HABAHAHAH. Thus (AH)-1AH. It follows that AH is a subgroupMathworldPlanetmathPlanetmath of G, and it is clearly finite.

For any xF we have x=r(xN)s(x)AB. So FAH, which completesPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath the proof.

Title local finiteness is closed under extensionPlanetmathPlanetmathPlanetmath, proof that
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