# Article

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Keywords:
group algebras; high subgroups; $p$-mixed and $p$-splitting groups; $\Sigma$-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.
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