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PROJECTIVE CHARACTERS WITH PRIME POWER DEGREES

Part of: Representation theory of groups

Published online by Cambridge University Press:  28 August 2018

YANG LIU Open the ORCID record for YANG LIU [Opens in a new window]
YANG LIU*
Affiliation:
College of Mathematical Science, Tianjin Normal University, Tianjin 300387, PR China email [email protected]
Rights & Permissions [Opens in a new window]

Abstract

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We consider the relationship between structural information of a finite group $G$ and $\text{cd}_{\unicode[STIX]{x1D6FC}}(G)$, the set of all irreducible projective character degrees of $G$ with factor set $\unicode[STIX]{x1D6FC}$. We show that for nontrivial $\unicode[STIX]{x1D6FC}$, if all numbers in $\text{cd}_{\unicode[STIX]{x1D6FC}}(G)$ are prime powers, then $G$ is solvable. Our result is proved by classical character theory using the bijection between irreducible projective representations and irreducible constituents of induced representations in its representation group.

Keywords

finite groupsprojective representationcharacter degrees

MSC classification

Primary: 20C15: Ordinary representations and characters
Secondary: 20C25: Projective representations and multipliers
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 99 , Issue 1 , February 2019 , pp. 78 - 82
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

Footnotes

The author is supported by NSFC of China (grant no. 11701421) and Introduction of talent research start-up fund of Tianjin Normal University (grant no. 5RL145).

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