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On groups with BFC-covered word values

Eloisa Detomi, Marta Morigi, Pavel Shumyatsky

Abstract

For a group G and a positive integer n write B_n(G) = {x \in G : |x^G | \le n}. If s is a positive integer and w is a group word, say that G satisfies the (n,s)-covering condition with respect to the word w if there exists a subset S of G such that |S| \le s and all w-values of G are contained in B_n(G)S. In a natural way, this condition emerged in the study of probabilistically nilpotent groups of class two. In this paper we obtain the following results. Let w be a multilinear commutator word on k variables and let G be a group satisfying the (n,s)-covering condition with respect to the word w. Then G has a soluble subgroup T such that the index [G : T] and the derived length of T are both (k,n,s)-bounded. Let G be a group satisfying the (n,s)-covering condition with respect to the word γ_k. Then (1) γ_{2k-1}(G) has a subgroup $T$ such that the index [γ_{2k-1}(G) : T] and |T'| are both (k,n,s)-bounded; and (2) G has a nilpotent subgroup U such that the index [G : U] and the nilpotency class of U are both (k,n,s)-bounded.

On groups with BFC-covered word values

Abstract

For a group G and a positive integer n write B_n(G) = {x \in G : |x^G | \le n}. If s is a positive integer and w is a group word, say that G satisfies the (n,s)-covering condition with respect to the word w if there exists a subset S of G such that |S| \le s and all w-values of G are contained in B_n(G)S. In a natural way, this condition emerged in the study of probabilistically nilpotent groups of class two. In this paper we obtain the following results. Let w be a multilinear commutator word on k variables and let G be a group satisfying the (n,s)-covering condition with respect to the word w. Then G has a soluble subgroup T such that the index [G : T] and the derived length of T are both (k,n,s)-bounded. Let G be a group satisfying the (n,s)-covering condition with respect to the word γ_k. Then (1) γ_{2k-1}(G) has a subgroup such that the index [γ_{2k-1}(G) : T] and |T'| are both (k,n,s)-bounded; and (2) G has a nilpotent subgroup U such that the index [G : U] and the nilpotency class of U are both (k,n,s)-bounded.
Paper Structure (4 sections, 48 equations)

This paper contains 4 sections, 48 equations.

Table of Contents

  1. Introduction
  2. A technical result
  3. Proof of Theorem \ref{['soluble-main']}
  4. Proof of Theorem \ref{['cov-nilp']}

Theorems & Definitions (9)

  • proof
  • proof
  • proof
  • proof : Proof of Proposition \ref{['pavels-lemma']}.
  • proof
  • proof
  • proof : Proof of Theorem \ref{['soluble-main']}
  • proof : Proof of Theorem \ref{['cov-nilp']} (1).
  • proof : Proof of Theorem \ref{['cov-nilp']} (2)