Journal of the Australian Mathematical Society

Table of Contents - 2001 - Volume 71, Issue 01  

Research Article

A problem of expressibility in some amalgamated products of groups

Valerii Faizieva1

a1 Tver Agricultural Academy Tver Russia e-mail: valeriy.fayziev@tversu.ru

Abstract

Let S be a subset of a group G such that S−1 = S. Denote by gr (S) the subgroup of G generated by S, and by ls(g) the length of an element g xs2208 gr(S) relative to the set S. Suppose that V is a finite subset of a free group F of countable rank such that the verbal subgroup V (F) is a proper subgroup of F. For an arbitrary group G, denote by S1446788700002743_inline1(G) the set of values in G of all the words from the set V. In the present paper, for amalgamated products G = A *H B such that A ≠ H and the number of double cosets of B by H is at least three, the infiniteness of the set {ls(g) | g xs2208 gr(S)}, where S = S1446788700002743_inline1(G) xs222A S1446788700002743_inline1(G)−1, is estabilished.

(Received February 01 1999)

(Revised November 13 2000)

2000 Mathematics subject classification

  • primary 20E06;
  • 20F22

Keywords and phrases

  • Group;
  • verbal subgroup;
  • the width of verbal subgroup