# Rendiconti del Seminario Matematico della Università di Padova

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**Volume 123, 2010, pp. 169–189**

**DOI: 10.4171/RSMUP/123-8**

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

Semra Öztürk Kaptanoglu^{[1]}(1) Department of Mathematics, Middle East Technical University, 06531, Ankara, Turkey

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, *C _{pn}*, 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*

_{2}is the unique maximal elementary abelian subgroup of

*C*

_{2}×

*C*

_{4}.

*No keywords available for this article.*

Öztürk Kaptanoglu Semra: Structure and Detection Theorems for $k[C_2\times C_4]$-Modules. *Rend. Sem. Mat. Univ. Padova* 123 (2010), 169-189. doi: 10.4171/RSMUP/123-8