15 - Growth sequences relative to subgroups

Summary

INTRODUCTION

Let G be a finite group. The object of this article is the comparison of the following growth sequences.

Γ(G,H). Let H be a subgroup of G. Denote by d(G,H) the minimal number of group elements needed to generate G together with H. For the direct product of n copies of a group X we write Xⁿ. The growth sequence Γ(G,H) of G relative to H is the sequence (d(Gⁿ,Hⁿ))_n∈ℕ.

Γ(I(G,H)). For a group X the augmentation ideal of its integral group ring ℤ G is denoted by I(X). The quotient I(G)/I(H)^↑G, where I(H)^↑G is the augmentation ideal of H induced to G, is denoted by I(G,H) and is called the augmentation ideal of G relative to H. The minimal number of generators of a ℤ G-module M is written as d_G(M), the direct product of n copies of M as Mⁿ. The growth sequence Γ(I(G,H)) of the relative augmentation ideal is the sequence (d_G(I(G,H)ⁿ)_n∈ℕ.

If H is the trivial subgroup of G, relative growth sequences reduce to ordinary ones. In this case we suppress the reference to H in all notation. Growth sequences Γ(G) were studied in, and, in the case of finitely generated groups in. It is clear that results for an arbitrary subgroup always contain the ordinary case and in this sense the results here extend those of the ordinary theory.