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Rendiconti del Seminario Matematico dell 'Universita' di Padova/Mathematical Journal of the University of Padova, vol.123, pp.169-189, 2010 (SCI-Expanded)
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, Cpn, or C2 × C2. Tackling the first interesting case, namely modules over k[C2 × C4], some structure theorems revealing the differences between elementary and non-elementary abelian group cases are obtained. The shifted cyclic subgroups of k[C2 × C4] are characterized. Using the direct sum decompositions of the restrictions of a k[C2 × C2]-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[C2 × C4]-modules having the same rank variety as k[C2 × C2]-modules can be detected by the set of multiplicities, where C2 × C2 is the unique maximal elementary abelian subgroup of C2 × C4.