# Structure and Detection Theorems for $k[C_2\times C_4]$-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 availableonly when *G* is cyclic, *Cpn*, 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]-modulescan be detected by the set of multiplicities,where _C_2×_C_2is the unique maximal elementary abelian subgroup of _C_2×_C_4.

## Cite this article

Semra Öztürk Kaptanoglu, Structure and Detection Theorems for $k[C_2\times C_4]$-Modules. Rend. Sem. Mat. Univ. Padova 123 (2010), pp. 169–189

DOI 10.4171/RSMUP/123-8