Growth sequences of finitely generated groups II

Abstract

A study is made of the minimum number of generators of the n-th direct power of certain finitely generated groups.

On the growth of generating sets for direct powers of semigroups

“…Wiegold and various co-authors in a series of papers investigated the sequence d(S), mostly when S is a finite group [14,15,16,17,8], but also when it is a finite semigroup [18] or an infinite group [19,13]. In the context of this paper we can summarise their main findings as follows.…”

Growth of Generating Sets for Direct Powers of Classical Algebraic Structures

“…In [23,30] Wiegold, with Stewart and Wilson respectively, investigates the sequence d(G) for (finitely generated) infinite groups G. The following main fundamental observation is that another type of growth-constant-makes its appearance.…”

“…This will be given a technical meaning here of being a group, ring, module, algebra, or Lie algebra. The d-sequences of groups have been investigated in considerable detail by Wiegold and various co-authors in a series of papers [18,23,[25][26][27][28]30]. Below we summarize their main findings.…”

A problem on growth sequences of groups

“…The aim of this paper is to consider Problem 1 posed by Stewart and Wiegold in [6]. The main result is that if G is a finitely generated perfect group having non-trivial finite images, then there exists a finite image B of G such that the growth sequence of B is eventuallly faster than that of every finite image of G. Moreover we investigate the growth sequences of simple groups of the same order.…”

“…The present article considers Problem 1 posed by Stewart and Wiegold in [6], as follows: 284 A. Erfanian [2] This is certainly true if G is not perfect. For, by the above remarks, d(G") = nd(G/G') for large n, in this case.…”