| Title:
|
Finite groups with a unique nonlinear nonfaithful irreducible character (English) |
| Author:
|
Iranmanesh, Ali |
| Author:
|
Saeidi, Amin |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
47 |
| Issue:
|
2 |
| Year:
|
2011 |
| Pages:
|
91-98 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper, we consider finite groups with precisely one nonlinear nonfaithful irreducible character. We show that the groups of order 16 with nilpotency class 3 are the only $p$-groups with this property. Moreover we completely characterize the nilpotent groups with this property. Also we show that if $G$ is a group with a nontrivial center which possesses precisely one nonlinear nonfaithful irreducible character then $G$ is solvable. (English) |
| Keyword:
|
minimal normal subgroups |
| Keyword:
|
faithful characters |
| Keyword:
|
strong condition on normal subgroups |
| Keyword:
|
Frobenius groups |
| MSC:
|
20C15 |
| MSC:
|
20D15 |
| MSC:
|
20F16 |
| idZBL:
|
Zbl 1249.20009 |
| idMR:
|
MR2813535 |
| . |
| Date available:
|
2011-06-06T14:38:56Z |
| Last updated:
|
2013-09-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141558 |
| . |
| Reference:
|
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| Reference:
|
[2] Di Martino, L., Tamburini, M. C.: Some remarks on the degrees of faithful irreducible representation of a finite group.Geom. Dedicata 41 (1992), 155–164. MR 1153979 |
| Reference:
|
[3] Fernández–Alcober, G. A., Moretó, A.: Groups with extreme character degrees and their normal subgroups.Trans. Amer. Math. Soc. 353 (2001), 2271–2292. 10.1090/S0002-9947-01-02685-X |
| Reference:
|
[4] Gagola, S. M.: A character theoretic condition for F(G)$>$1.Comm. Algebra 133 (2005), 1369–1382. Zbl 1098.20005, MR 2149064, 10.1081/AGB-200058414 |
| Reference:
|
[5]
: GAP Groups, Algorithms, and Programming, Version 4.4.10, 2007. |
| Reference:
|
[6] Isaacs, I. M.: Character Theory of Finite Groups.Dover, New York, 1994. Zbl 0849.20004, MR 1280461 |
| Reference:
|
[7] Loukaki, M.: On distinct character degrees.Israel J. Math. 159 (2007), 93–97. Zbl 1131.20006, MR 2342474, 10.1007/s11856-007-0039-1 |
| Reference:
|
[8] Seitz, G. M.: Finite groups having only one irreducible representation of degree greater than one.Proc. Amer. Math. Soc. 19 (1968), 459–461. Zbl 0244.20010, MR 0222160, 10.1090/S0002-9939-1968-0222160-X |
| . |
