Isomorphism of Modular Group Algebras ofp-Mixed Abelian Groups

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 Danchev , P. V. ( 1998 ). Isomorphism of commutative group algebras of mixed splitting groups . Compt. Rend. Acad. Bulg. Sci. 51 : 13 – 16 . Danchev , P. V. ( 2000 ). Isomorphism of modular group algebras of totally projective abelian groups . Communications in Algebra 28 : 2521 – 2531 . Danchev , P. V. ( 2001 ). On a question of W. L. May concerning the isomorphism of modular group algebras . Communications in Algebra 29 : 1953 – 1958 . Danchev , P. V. ( 2001 ). Normed units in Abelian group rings . Glasg. Math. J. 43 : 365 – 373 . Danchev , P. V. ( 2002 ). Invariants for group algebras of splitting abelian groups with simply presented components . Compt. Rend. Acad. Bulg. Sci. 55 : 5 – 8 . Danchev , P. V. ( 2004 ). A note on the isomorphic modular group algebras of abelian groups with simply presented p-components . Compt. Rend. Acad. Bulg. Sci. 57 : 13 – 14 . ].