| Title:
|
Commutative modular group algebras of $p$-mixed and $p$-splitting abelian $\Sigma$-groups (English) |
| Author:
|
Danchev, Peter |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
43 |
| Issue:
|
3 |
| Year:
|
2002 |
| Pages:
|
419-428 |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
group algebras |
| Keyword:
|
high subgroups |
| Keyword:
|
$p$-mixed and $p$-splitting groups |
| Keyword:
|
$\Sigma $-groups |
| MSC:
|
16S34 |
| MSC:
|
16U60 |
| MSC:
|
20C07 |
| MSC:
|
20K10 |
| MSC:
|
20K20 |
| MSC:
|
20K21 |
| idZBL:
|
Zbl 1068.16042 |
| idMR:
|
MR1920518 |
| . |
| Date available:
|
2009-01-08T19:23:27Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119332 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
[3] Danchev P.V.: Commutative group algebras of abelian $\Sigma$-groups.Math. J. Okayama Univ. 40 (1998), 77-90. MR 1755921 |
| Reference:
|
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| Reference:
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| Reference:
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| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
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| . |
