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Degree Homogeneous Subgroups

Published online by Cambridge University Press:  20 November 2018

John D. Dixon  and
A. Rahnamai Barghi
John D. Dixon
Affiliation:
School of Mathematics and Statistics, Carleton University, Ottawa, ON, K1S 5B6 e-mail: [email protected]
A. Rahnamai Barghi
Affiliation:
Institute for Advanced Studies in Basic Sciences, P.O. Box 45195-159, Zanjan, Iran e-mail: [email protected]
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Abstract

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Let $G$ be a finite group and $H$ be a subgroup. We say that $H$ is degree homogeneous if, for each $x\,\in \,\text{Irr}\left( G \right)$, all the irreducible constituents of the restriction ${{x}_{H}}$ have the same degree. Subgroups which are either normal or abelian are obvious examples of degree homogeneous subgroups. Following a question by E.M. Zhmud’, we investigate general properties of such subgroups. It appears unlikely that degree homogeneous subgroups can be characterized entirely by abstract group properties, but we providemixed criteria (involving both group structure and character properties) which are both necessary and sufficient. For example, $H$ is degree homogeneous in $G$ if and only if the derived subgroup ${H}'$ is normal in $G$ and, for every pair $\alpha $, $\beta $ of irreducible $G$-conjugate characters of ${H}'$, all irreducible constituents of ${{\alpha }^{H}}$ and ${{\beta }^{H}}$ have the same degree.

Keywords

20C15
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 48 , Issue 1 , 01 March 2005 , pp. 41 - 49
Copyright
Copyright © Canadian Mathematical Society 2005

References

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