# Structure and Detection Theorems for $k[C_{2}×C_{4}]$-Modules

### Semra Öztürk Kaptanoglu

Middle East Technical University, Ankara, Turkey

## 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 $C_{2}×C_{2}$. Tackling the first interesting case, namely modules over $k[C_{2}×C_{4}]$, some structure theorems revealing the differences between elementary and non-elementary abelian group cases are obtained. The shifted cyclic subgroups of $k[C_{2}×C_{4}]$ are characterized. Using the direct sum decompositions of the restrictions of a $k[C_{2}×C_{2}]$-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 $k[C_{2}×C_{4}]$-modules having the same rank variety as $k[C_{2}×C_{2}]$-modules can be detected by the set of multiplicities, where $C_{2}×C_{2}$ is the unique maximal elementary abelian subgroup of $C_{2}×C_{4}$.

## Cite this article

Semra Öztürk Kaptanoglu, Structure and Detection Theorems for $k[C_{2}×C_{4}]$-Modules. Rend. Sem. Mat. Univ. Padova 123 (2010), pp. 169–189

DOI 10.4171/RSMUP/123-8