Abstract
Let G be a fixed finite group and consider a short exact sequence: 1 + s →E →G +1, where E is a finitely generated group. The abelian group 3 Ŝ= S/S' where may be regarded as a ZG-module and, for a fixed prime p, the elementary = Ŝ/S'sp= Ŝ/pŜmay be regarded as an IFpG-module. abelian p-group If %3! E is a free group, is called the relation module of G determined by Ψ and s the relation module modulo p. In general we call Ŝ the relative relation module, and S the relative relation module modulo p . When the minimal number of generators of G and E is the same, and 3 will be called minimal. Gaschütz, Gruenberg and others have studied relation modules and relation modules modulo p. The main aim of this thesis is to study relative relation modules modulo p when E is a free product of cyclic To be more precise, let X = {gi,1 ≤1i≤d} be a generating set groups.of G, Gi, the cyclic group generated by 9; E the free product of the Gi1 d, v the epimorphism whose restriction to each G, is the identity isomorphism, and Ŝ the kernel of Ψ. Some of the results may be summarised as follows. o f the I is embedded in the direct s um of the augmentation ideals of the IFp Gisisd, induced to G, and the resulting factor module is isomorphic to the augmentation ideal of IFp G. Ŝ may also be embedded in a free IF pG-module of rank d -1. Two relative relation modules, isomorphic as iFp-spaces, are rarely isomorphic as G-modules; that is, Ŝ not only depends on G, p and d but also on ѱ . Some cases when Ŝ does not depend on ip are established.