  Let G be a group of the form G_1* ... *G_n, the free product of n subgroups,
and let M be a ZG-module of the form $\bigoplus_{i=1}^n M_i
\otimes_{\mathbb{Z}G_i} \mathbb{Z}G$. We shall give formulae in various
situations for $d_{ZG}(M)$, the minimum number of elements required to generate
M. In particular if C_1,C_2 are non-trivial finite cyclic groups of coprime
orders, $G = (C_1 \times Z) * (C_2 \times Z)$ and $F/R \cong G$ is the free
presentation obtained from the natural free presentations of the two factors,
then the number of generators of the relation module, $d_{\mathbb{Z}G}(R/R')$
is three. It seems plausible that the minimum number of relators of G should be
4, and this would give a finitely presented group with positive relation gap.
However we cannot prove this last statement.
