Generators and commutators in finite groups; abstract quotients of compact groups

TL;DR

This paper investigates the structure of subgroups in finite and compact groups, providing new bounds and proofs for properties of quotients and commutators, with implications for profinite and topological groups.

Contribution

It introduces bounds on commutator products in finite groups and applies these results to properties of quotients in profinite and compact groups.

Findings

G^n is closed and open in finitely generated profinite groups

Finitely generated abstract quotients of compact groups are finite

Existence of infinite quotients linked to virtually abelian quotients in compact groups

Abstract

Let N be a normal subgroup of a finite group G. We prove that under certain (unavoidable) conditions the subgroup [N,G] is a product of commutators [N,y] (with prescribed values of y from a given set Y) of length bounded by a function of d(G) and |Y| only. This has several applications: 1. A new proof that G^n is closed (and hence open) in any finitely generated profinite group G. 2. A finitely generated abstract quotient of a compact Hausdorff group must be finite. 3. Let G be a topologically finitely generated compact Hausdorff group. Then G has a countably infinite abstract quotient if and only if G has an infinite virtually abelian continuous quotient.