Finite groups with a unique nonlinear nonfaithful irreducible character

Ali Iranmanesh; Amin Saeidi

Archivum Mathematicum (2011)

  • Volume: 047, Issue: 2, page 91-98
  • ISSN: 0044-8753

Abstract

top
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.

How to cite

top

Iranmanesh, Ali, and Saeidi, Amin. "Finite groups with a unique nonlinear nonfaithful irreducible character." Archivum Mathematicum 047.2 (2011): 91-98. <http://eudml.org/doc/116537>.

@article{Iranmanesh2011,
abstract = {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.},
author = {Iranmanesh, Ali, Saeidi, Amin},
journal = {Archivum Mathematicum},
keywords = {minimal normal subgroups; faithful characters; strong condition on normal subgroups; Frobenius groups; nonlinear irreducible characters; finite groups; minimal normal subgroups; faithful characters; Frobenius groups},
language = {eng},
number = {2},
pages = {91-98},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Finite groups with a unique nonlinear nonfaithful irreducible character},
url = {http://eudml.org/doc/116537},
volume = {047},
year = {2011},
}

TY - JOUR
AU - Iranmanesh, Ali
AU - Saeidi, Amin
TI - Finite groups with a unique nonlinear nonfaithful irreducible character
JO - Archivum Mathematicum
PY - 2011
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 047
IS - 2
SP - 91
EP - 98
AB - 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.
LA - eng
KW - minimal normal subgroups; faithful characters; strong condition on normal subgroups; Frobenius groups; nonlinear irreducible characters; finite groups; minimal normal subgroups; faithful characters; Frobenius groups
UR - http://eudml.org/doc/116537
ER -

References

top
  1. Berkovich, Y., Chillag, D., Herzog, M., 10.1090/S0002-9939-1992-1088438-9, Proc. Amer. Math. Soc. 115 (1992), 955–958. (1992) Zbl0822.20004MR1088438DOI10.1090/S0002-9939-1992-1088438-9
  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. (1992) MR1153979
  3. Fernández–Alcober, G. A., Moretó, A., 10.1090/S0002-9947-01-02685-X, Trans. Amer. Math. Soc. 353 (2001), 2271–2292. (2001) DOI10.1090/S0002-9947-01-02685-X
  4. Gagola, S. M., 10.1081/AGB-200058414, Comm. Algebra 133 (2005), 1369–1382. (2005) Zbl1098.20005MR2149064DOI10.1081/AGB-200058414
  5. GAP Groups, Algorithms, and Programming, Version 4.4.10, 2007 
  6. Isaacs, I. M., Character Theory of Finite Groups, Dover, New York, 1994. (1994) Zbl0849.20004MR1280461
  7. Loukaki, M., 10.1007/s11856-007-0039-1, Israel J. Math. 159 (2007), 93–97. (2007) Zbl1131.20006MR2342474DOI10.1007/s11856-007-0039-1
  8. Seitz, G. M., 10.1090/S0002-9939-1968-0222160-X, Proc. Amer. Math. Soc. 19 (1968), 459–461. (1968) Zbl0244.20010MR0222160DOI10.1090/S0002-9939-1968-0222160-X

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.