Módulos de permutação p-ádicos para p-grupos abelianos elementares

Abstract

Let $\Z_p$ be the ring of $p$-adic integers and $G$ be a finite $p$-group. Recently, MacQuarrie and Zalesskii characterized the $\Z_pG$-permutation modules by just looking at modules for $G/N$, where $N$ is a normal subgroup of $G$ with order $p$. This characterization is given by two conditions and in this work we show that, in general, we cannot remove either of these conditions to characterize the permutation $\Z_pG$-modules. The authors already knew that one of the conditions could not be removed but the necessity of the other condition was unknown. We work with a correspondence due to Butler to construct a $\Z_pG$-module that is not a $\Z_pG$-permutation module, but which satisfies the condition that might still have been a characterization of permutation $\Z_pG$-modules.