| Title:
|
Basic subgroups in modular abelian group algebras (English) |
| Author:
|
Danchev, Peter |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
57 |
| Issue:
|
1 |
| Year:
|
2007 |
| Pages:
|
173-182 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Suppose ${F}$ is a perfect field of ${\mathop {\mathrm char}F=p\ne 0}$ and ${G}$ is an arbitrary abelian multiplicative group with a ${p}$-basic subgroup ${B}$ and ${p}$-component ${G_p}$. Let ${FG}$ be the group algebra with normed group of all units ${V(FG)}$ and its Sylow ${p}$-subgroup ${S(FG)}$, and let ${I_p(FG;B)}$ be the nilradical of the relative augmentation ideal ${I(FG;B)}$ of ${FG}$ with respect to ${B}$. The main results that motivate this article are that ${1+I_p(FG;B)}$ is basic in ${S(FG)}$, and ${B(1+I_p(FG;B))}$ is ${p}$-basic in ${V(FG)}$ provided ${G}$ is ${p}$-mixed. These achievements extend in some way a result of N. Nachev (1996) in Houston J. Math. when ${G}$ is $p$-primary. Thus the problem of obtaining a ($p$-)basic subgroup in ${FG}$ is completely resolved provided that the field $F$ is perfect. Moreover, it is shown that ${G_p(1+I_p(FG;B))/G_p}$ is basic in ${S(FG)/ G_p}$, and $G(1+I_p(FG; B))/G$ is basic in ${V(FG)/G}$ provided ${G}$ is ${p}$-mixed. As consequences, ${S(FG)}$ and ${S(FG)/G_p}$ are both starred or divisible groups. All of the listed assertions enlarge in a new aspect affirmations established by us in Czechoslovak Math. J. (2002), Math. Bohemica (2004) and Math. Slovaca (2005) as well. (English) |
| Keyword:
|
$p$-basic subgroups |
| Keyword:
|
normalized units |
| Keyword:
|
group algebras |
| Keyword:
|
starred groups |
| MSC:
|
16S34 |
| MSC:
|
16U60 |
| MSC:
|
20K10 |
| MSC:
|
20K20 |
| MSC:
|
20K21 |
| idZBL:
|
Zbl 1167.16026 |
| idMR:
|
MR2309958 |
| . |
| Date available:
|
2011-01-04T15:35:17Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/128164 |
| . |
| 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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| Reference:
|
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| Reference:
|
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| . |
