A problem on growth sequences of groups

Abstract

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.

A note on growth sequences of alternating groups

“…Since r(n) is a decreasing function for n м 3 by Lemma 2.3, therefore g b (n) < 2 3 2 r(n) Ϲ 2 3 2 r(48) < 1 6 for n м 48. Hence for all n м 48, g b (n) < 1 6 .…”

“…The growth sequence of finite groups are known with accuracy in terms of various parameters [9][10][11][12], and quite a lot is known in the case of finitely generated infinite groups [3,13]. One of the main theoretical tools in the finite case is a result of P. Hall [4], showing that for a finite non-abelian simple group G, and k м d(G),…”

Growth Rates of Algebras, I: Pointed Cube Terms

“…Growth rates of groups, semigroups and group expansions. Wiegold's paper initiated a program of research into growth rates of groups including, for example, [5,6,7,8,9,10,11,22,23,26,27,31,33,34,35,37,38]. The program expanded to include the investigation of growth rates of semigroups, in [28,36], and later to include the investigation of growth rates of more general algebraic structures, in [14,30].…”