Isomorphism of Modular Group Algebras of p-Mixed Abelian Groups

Abstract

Let A be a torsion-free abelian group and F a free subgroup of A. We prove that if A/F is a reduced p-group and A/(F + C) is reduced for every p-pure subgroup C of A, then A is free. Let KG be the group algebra of an abelian group G over a field K of prime characteristic p. Denote by S(KG) the p-component of the group V(KG) of normalized units of KG (of augmentation 1). Let H be an arbitrary group and KH ≅ KG as K-algebras. We prove the following. First, assume that G is a splitting group, the p-component G p of G is simply presented, and the field K is perfect. Then H p ≅ G p . If, in addition, G is p-mixed, then G p is a direct factor of S(KG), and G is a direct factor of V(KG), each with the same simply presented complement. Secondly, we introduce a class of special p-mixed abelian groups and prove that, if G belong to this class, then any group basis of the group algebra KG splits. Besides, H is p-mixed and splits. Thirdly, if G is a special p-mixed abelian group and G p is a reduced totally projective p-group, then H ≅ G. These results correct some essential inaccuracies and incompleteness in the proofs of results in this direction of Danchev [3-8].