On profinite groups with commutators covered by countably many cosets

TL;DR

This paper investigates the structure of verbal subgroups in profinite groups when the set of certain word-values is covered by countably many cosets, establishing conditions under which these subgroups are virtually within a specific class of groups.

Contribution

It proves that under certain conditions, the verbal subgroup generated by multilinear commutators is virtually within a class of groups closed under subgroups, quotients, and finite products of normal subgroups.

Findings

Verbal subgroup w(G) is virtually-C under given conditions.

Strengthens existing results on profinite groups and commutator coverage.

Provides conditions for the structure of w(G) based on coset coverings.

Abstract

Let w be a group-word. Suppose that the set of all w-values in a profinite group G is contained in a union of countably many cosets of subgroups. We are concerned with the question to what extent the structure of the verbal subgroup w(G) depends on the properties of the subgroups. We prove the following theorem. Let C be a class of groups closed under taking subgroups, quotients, and such that in any group the product of finitely many normal C-subgroups is again a C-subgroup. If w is a multilinear commutator and G is a profinite group such that the set of all w-values is contained in a union of countably many cosets g_iG_i where each G_i is in C, then the verbal subgroup w(G) is virtually-C. This strengthens several known results.