Structure and Detection Theorems for $k[C_2\times C_4]$-Modules

Abstract

Let $k[G]$ be the group algebra, where $G$ is a finite abelian $p$-group and $k$ is a field of characteristic $p$. A complete classification of finitely generated $k[G]$-modules is available only when $G$ is cyclic, $C_{p^n}$, or $C_2\times C_2$. Tackling the first interesting case, namely modules over $k[C_2\times C_4]$, some structure theorems revealing the differences between elementary and non-elementary abelian group cases are obtained. The shifted cyclic subgroups of $k[C_2\times C_4]$ are characterized. Using the direct sum decompositions of the restrictions of a $k[C_2\times C_2]$-module $M$ to shifted cyclic subgroups we define the set of multiplicities of $M$. It is an invariant richer than the rank variety. Certain types of $k[C_2\times C_4]$-modules having the same rank variety as $k[C_2\times C_2]$-modules can be detected by the set of multiplicities, where $C_2\times C_2$ is the unique maximal elementary abelian subgroup of $C_2\times C_4$.