| Title:
|
Isomorphism of commutative group algebras of $p$-mixed splitting groups over rings of characteristic zero (English) |
| Author:
|
Danchev, Peter |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
131 |
| Issue:
|
1 |
| Year:
|
2006 |
| Pages:
|
85-93 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Suppose $G$ is a $p$-mixed splitting abelian group and $R$ is a commutative unitary ring of zero characteristic such that the prime number $p$ satisfies $p\notin \mathop {\text{inv}}(R) \cup \mathop {\text{zd}}(R)$. Then $R(H)$ and $R(G)$ are canonically isomorphic $R$-group algebras for any group $H$ precisely when $H$ and $G$ are isomorphic groups. This statement strengthens results due to W. May published in J. Algebra (1976) and to W. Ullery published in Commun. Algebra (1986), Rocky Mt. J. Math. (1992) and Comment. Math. Univ. Carol. (1995). (English) |
| Keyword:
|
group algebras |
| Keyword:
|
isomorphisms |
| Keyword:
|
$p$-mixed splitting groups |
| Keyword:
|
rings with zero characteristic |
| MSC:
|
16S34 |
| MSC:
|
16U60 |
| MSC:
|
20C07 |
| MSC:
|
20K10 |
| MSC:
|
20K21 |
| idZBL:
|
Zbl 1111.20007 |
| idMR:
|
MR2211005 |
| DOI:
|
10.21136/MB.2006.134084 |
| . |
| Date available:
|
2009-09-24T22:24:28Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134084 |
| . |
| Reference:
|
[1] D. Beers, F. Richman, E. A. Walker: Group algebras of abelian groups.Rend. Sem. Mat. Univ. Padova 69 (1983), 41–50. MR 0716984 |
| Reference:
|
[2] P. V. Danchev: Isomorphic semisimple group algebras.C. R. Acad. Bulg. Sci. 53 (2000), 13–14. Zbl 0964.20001, MR 1779521 |
| Reference:
|
[3] P. V. Danchev: A new simple proof of the W. May’s claim: $FG$ determines $G/G_0$.Riv. Mat. Univ. Parma 1 (2002), 69–71. Zbl 1019.20002, MR 1951976 |
| Reference:
|
[4] P. V. Danchev: A note on isomorphic commutative group algebras over certain rings.An. St. Univ. Ovidius Constanta 13 (2005), 69–74. Zbl 1113.20005, MR 2230861 |
| Reference:
|
[5] J. M. Irwin, S. A. Khabbaz, G. Rayna: The role of the tensor product in the splitting of abelian groups.J. Algebra 14 (1970), 423–442. MR 0255675, 10.1016/0021-8693(70)90093-1 |
| Reference:
|
[6] G. Karpilovsky: On some properties of group rings.J. Austral. Math. Soc., Ser. A 29 (1980), 385–392. Zbl 0432.16007, MR 0578697, 10.1017/S1446788700021534 |
| Reference:
|
[7] W. L. May: Commutative group algebras.Trans. Amer. Math. Soc. 136 (1969), 139–149. Zbl 0182.04401, MR 0233903, 10.1090/S0002-9947-1969-0233903-9 |
| Reference:
|
[8] W. L. May: Invariants for commutative group algebras, Ill.J. Math. 15 (1971), 525–531. MR 0286903, 10.1215/ijm/1256052619 |
| Reference:
|
[9] W. L. May: Group algebras over finitely generated rings.J. Algebra 39 (1976), 483–511. Zbl 0328.16012, MR 0399232, 10.1016/0021-8693(76)90049-1 |
| Reference:
|
[10] W. L. May: Isomorphism of group algebras.J. Algebra 40 (1976), 10–18. Zbl 0329.20002, MR 0414618, 10.1016/0021-8693(76)90083-1 |
| Reference:
|
[11] W. D. Ullery: Isomorphism of group algebras.Commun. Algebra 14 (1986), 767–785. Zbl 0587.16011, MR 0834462, 10.1080/00927878608823334 |
| Reference:
|
[12] W. D. Ullery: On isomorphism of group algebras of torsion abelian groups.Rocky Mt. J. Math. 22 (1992), 1111–1122. Zbl 0773.16008, MR 1183707, 10.1216/rmjm/1181072715 |
| Reference:
|
[13] W. D. Ullery: A note on group algebras of $p$-primary abelian groups.Comment. Math. Univ. Carol. 36 (1995), 11–14. Zbl 0828.20005, MR 1334408 |
| . |
