Commutative modular group algebras of $p$-mixed and $p$-splitting abelian $\Sigma $-groups

summary:Let $G$ be a $p$-mixed abelian group and $R$ is a commutative perfect integral domain of $\operatorname{char} R = p > 0$. Then, the first main result is that the group of all normalized invertible elements $V(RG)$ is a $\Sigma $-group if and only if $G$ is a $\Sigma $-group. In particular, the second central result is that if $G$ is a $\Sigma $-group, the $R$-algebras isomorphism $RA\cong RG$ between the group algebras $RA$ and $RG$ for an arbitrary but fixed group $A$ implies $A$ is a $p$-mixed abelian $\Sigma $-group and even more that the high subgroups of $A$ and $G$ are isomorphic, namely, ${\Cal H}_A \cong {\Cal H}_G$. Besides, when $G$ is $p$-splitting and $R$ is an algebraically closed field of $\operatorname{char} R = p \not= 0$, $V(RG)$ is a $\Sigma $-group if and only if $G_p$ and $G/G_t$ are both $\Sigma $-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.