Minimal relative relation modules of finite $p$-groups

Abstract:

Consider $1 \to S \to E \to G \to 1$, where $G$ is a finite $p$-group generated by ${g_i},\;1 \leqslant i \leqslant d$, and $E$ a free product of cyclic groups $\langle {g_i}\rangle ,1 \leqslant i \leqslant d$. If $d$ is the minimum number of generators for $G$, then we prove that the largest elementary abelian $p$-quotient $S/{S’}{S^p}$, regarded as an ${\mathbb {F}_p}G$-module via conjugation in $E$, is nonprojective and indecomposable.