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 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.