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

Commentationes Mathematicae Universitatis Carolinae (2002)

- Volume: 43, Issue: 3, page 419-428
- ISSN: 0010-2628

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topDanchev, Peter Vassilev. "Commutative modular group algebras of $p$-mixed and $p$-splitting abelian $\Sigma $-groups." Commentationes Mathematicae Universitatis Carolinae 43.3 (2002): 419-428. <http://eudml.org/doc/248994>.

@article{Danchev2002,

abstract = {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, $\{\mathcal \{H\}\}_A \cong \{\mathcal \{H\}\}_G$. Besides, when $G$ is $p$-splitting and $R$ is an algebraically closed field of $\operatorname\{char\} R = p \ne 0$, $V(RG)$ 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.},

author = {Danchev, Peter Vassilev},

journal = {Commentationes Mathematicae Universitatis Carolinae},

keywords = {group algebras; high subgroups; $p$-mixed and $p$-splitting groups; $\Sigma $-groups; group algebras; high subgroups; -mixed Abelian groups; -splitting Abelian groups; Abelian -groups; groups of units; isomorphism problem},

language = {eng},

number = {3},

pages = {419-428},

publisher = {Charles University in Prague, Faculty of Mathematics and Physics},

title = {Commutative modular group algebras of $p$-mixed and $p$-splitting abelian $\Sigma $-groups},

url = {http://eudml.org/doc/248994},

volume = {43},

year = {2002},

}

TY - JOUR

AU - Danchev, Peter Vassilev

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

JO - Commentationes Mathematicae Universitatis Carolinae

PY - 2002

PB - Charles University in Prague, Faculty of Mathematics and Physics

VL - 43

IS - 3

SP - 419

EP - 428

AB - 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, ${\mathcal {H}}_A \cong {\mathcal {H}}_G$. Besides, when $G$ is $p$-splitting and $R$ is an algebraically closed field of $\operatorname{char} R = p \ne 0$, $V(RG)$ 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.

LA - eng

KW - group algebras; high subgroups; $p$-mixed and $p$-splitting groups; $\Sigma $-groups; group algebras; high subgroups; -mixed Abelian groups; -splitting Abelian groups; Abelian -groups; groups of units; isomorphism problem

UR - http://eudml.org/doc/248994

ER -

## References

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