Extending Huppert’s conjecture from non-abelian simple groups to quasi-simple groups. (English) Zbl 1372.20015

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

Ill. J. Math. 59, No. 4, 901-924 (2015).

Summary: We propose to extend a conjecture of B. Huppert [ibid. 44, No. 4, 828–842 (2000; Zbl 0972.20006)] 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 \(3\).

MSC:

20C15	Ordinary representations and characters
20C33	Representations of finite groups of Lie type
20C34	Representations of sporadic groups
20C30	Representations of finite symmetric groups
20D60	Arithmetic and combinatorial problems involving abstract finite groups

Keywords:

Huppert’s conjecture; irreducible characters; finite quasi-simple groups; set of character degrees; special linear group

Citations:

Zbl 0972.20006