Commutative modular group algebras of p-mixed and p-splitting abelian Σ-groups

Let G be a p-mixed abelian group and R is a commutative perfect integral domain of charR = p > 0. Then, the first main result is that the group of all normalized invertible elements V (RG) is a Σ-group if and only if G is a Σ-group. In particular, the second central result is that if G is a Σ-group, the R-algebras isomorphism RA ∼= RG between the group algebras RA and RG for an arbitrary but fixed group A implies A is a p-mixed abelian Σ-group and even more that the high subgroups of A and G are isomorphic, namely, HA ∼= HG. Besides, when G is p-splitting and R is an algebraically closed field of charR = p 6= 0, V (RG) is a Σ-group if and only if Gp and G/Gt are both Σ-groups. These statements combined with our recent results published in Math. J. Okayama Univ. (1998) almost exhausted the investigations on this theme concerning the description of the group structure.