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Jordan type of a k[C(p)xC(p)]-module

Let E be the elementary abelian group C(p)xC(p), k a field of characteristic p, M a finite dimensional module over the group algebra k[E] and J the Jacobson radical J of k[E]. We prove that the decomposition of M when considered as a k[]-module for a p-point x in J is well defined modulo J(p).