Geodesic
Commutative modular group algebras of $p$-mixed and $p$-splitting abelian $\Sigma$-groups
Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) no. 3, pp. 419-428

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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.
Classification : 16S34, 16U60, 20C07, 20K10, 20K20, 20K21
Keywords: group algebras; high subgroups; $p$-mixed and $p$-splitting groups; $\Sigma $-groups 
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     title = {Commutative modular group algebras of $p$-mixed and $p$-splitting abelian $\Sigma$-groups},
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Danchev, Peter. Commutative modular group algebras of $p$-mixed and $p$-splitting abelian $\Sigma$-groups. Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) no. 3, pp. 419-428. http://geodesic.mathdoc.fr/item/CMUC_2002__43_3_a2/