Minimal Relations for Certain Wreath Products of Groups

Abstract

Let p be a rational prime, G a non-trivial finite p group, and K the field of p elements, regarded as a trivial G-module according to context; then we define:d(G) = dimKH1(G, K), the minimal number of generators of G,r(G) = dimKH2(G, K),r′(G) = the minimal number of relations required to define G,where, in the last equation, it is sufficient to take the minimum over those presentations of G with d(G) generators. It is well known (see § 2) that the following inequalities hold:We shall consider only finite p-groups, so that the class of groups with r = d coincides with that consisting of those groups whose Schur multiplicator is trivial.