Illinois Journal of Mathematics

Extending Huppert’s conjecture from non-Abelian simple groups to quasi-simple groups

Nguyen Ngoc Hung, Philani R. Majozi, Hung P. Tong-Viet, and Thomas P. Wakefield

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Abstract

We propose to extend a conjecture of Bertram Huppert [Illinois J. Math. 44 (2000) 828–842] from finite non-Abelian simple groups to finite quasi-simple groups. Specifically, we conjecture that if a finite group $G$ and a finite quasi-simple group $H$ with ${\mathrm{Mult}}(H/\mathbf{Z}(H))$ cyclic have the same set of irreducible character degrees (not counting multiplicity), then $G$ is isomorphic to a central product of $H$ and an Abelian group. We present a pattern to approach this extended conjecture and, as a demonstration, we confirm it for the special linear groups in dimensions $2$ and $3$.

Article information

Source
Illinois J. Math., Volume 59, Number 4 (2015), 901-924.

Dates
Received: 9 November 2016
First available in Project Euclid: 27 February 2017

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1488186014

Digital Object Identifier
doi:10.1215/ijm/1488186014

Mathematical Reviews number (MathSciNet)
MR3628294

Zentralblatt MATH identifier
1372.20015

Subjects
Primary: 20C15: Ordinary representations and characters
Secondary: 20C33: Representations of finite groups of Lie type 20C34: Representations of sporadic groups 20C30: Representations of finite symmetric groups

Citation

Hung, Nguyen Ngoc; Majozi, Philani R.; Tong-Viet, Hung P.; Wakefield, Thomas P. Extending Huppert’s conjecture from non-Abelian simple groups to quasi-simple groups. Illinois J. Math. 59 (2015), no. 4, 901--924. doi:10.1215/ijm/1488186014. https://projecteuclid.org/euclid.ijm/1488186014


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