Commutative modular group algebras of $p$-mixed and $p$-splitting abelian $\Sigma$-groups
Danchev, Peter Vassilev
Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002), p. 419-428 / Harvested from Czech Digital Mathematics Library
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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.

Publié le : 2002-01-01
Classification:  16S34,  16U60,  20C07,  20K10,  20K20,  20K21 
@article{119332,
     author = {Peter Vassilev Danchev},
     title = {Commutative modular group algebras of $p$-mixed and $p$-splitting abelian $\Sigma$-groups},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {43},
     year = {2002},
     pages = {419-428},
     zbl = {1068.16042},
     mrnumber = {1920518},
     language = {en},
     url = {http://dml.mathdoc.fr/item/119332}
}

Danchev, Peter Vassilev. Commutative modular group algebras of $p$-mixed and $p$-splitting abelian $\Sigma$-groups. Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) pp. 419-428. http://gdmltest.u-ga.fr/item/119332/

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