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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Groups with two extreme character degrees and their normal subgroups


Authors: Gustavo A. Fernández-Alcober and Alexander Moretó
Journal: Trans. Amer. Math. Soc. 353 (2001), 2171-2192
MSC (2000): Primary 20C15, 20D15
Published electronically: February 7, 2001
MathSciNet review: 1814066
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Abstract:

We study the finite groups $G$ for which the set $\operatorname{cd}(G)$ of irreducible complex character degrees consists of the two most extreme possible values, that is, $1$ and $\vert G:Z(G)\vert^{1/2}$. We are easily reduced to finite $p$-groups, for which we derive the following group theoretical characterization: they are the $p$-groups such that $\vert G:Z(G)\vert$ is a square and whose only normal subgroups are those containing $G'$ or contained in $Z(G)$. By analogy, we also deal with $p$-groups such that $\vert G:Z(G)\vert=p^{2n+1}$ is not a square, and we prove that $\operatorname{cd}(G) =\{1,p^n\}$ if and only if a similar property holds: for any $N\trianglelefteq G$, either $G'\le N$ or $\vert NZ(G):Z(G)\vert\le p$. The proof of these results requires a detailed analysis of the structure of the $p$-groups with any of the conditions above on normal subgroups, which is interesting for its own sake. It is especially remarkable that these groups have small nilpotency class and that, if the nilpotency class is greater than $2$, then the index of the centre is small, and in some cases we may even bound the order of $G$.


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Additional Information

Gustavo A. Fernández-Alcober
Affiliation: Departamento de Matemáticas, Universidad del País Vasco, 48080 Bilbao (Spain)
Email: mtpfealg@lg.ehu.es

Alexander Moretó
Affiliation: Departamento de Matemáticas, Universidad del País Vasco, 48080 Bilbao (Spain)
Email: mtbmoqua@lg.ehu.es

DOI: http://dx.doi.org/10.1090/S0002-9947-01-02685-X
PII: S 0002-9947(01)02685-X
Received by editor(s): June 9, 1999
Received by editor(s) in revised form: December 8, 1999
Published electronically: February 7, 2001
Additional Notes: Research of the second author supported by a grant of the Basque Government and by the University of the Basque Country grant UPV 127.310-EB160/98.
Article copyright: © Copyright 2001 American Mathematical Society