Vol. 9, No. 3, 2015

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Complex group algebras of the double covers of the symmetric and alternating groups

Christine Bessenrodt, Hung Ngoc Nguyen, Jørn B. Olsson and Hung P. Tong-Viet

Vol. 9 (2015), No. 3, 601–628

We prove that the double covers of the alternating and symmetric groups are determined by their complex group algebras. To be more precise, let n 5 be an integer, G a finite group, and let Ân and Ŝn± denote the double covers of An and Sn, respectively. We prove that GÂn if and only if GÂn, and GŜn+Ŝn if and only if GŜn+ or Ŝn. This in particular completes the proof of a conjecture proposed by the second and fourth authors that every finite quasisimple group is determined uniquely up to isomorphism by the structure of its complex group algebra. The known results on prime power degrees and relatively small degrees of irreducible (linear and projective) representations of the symmetric and alternating groups together with the classification of finite simple groups play an essential role in the proofs.

symmetric groups, alternating groups, complex group algebras, Schur covers, double covers, irreducible representations, character degrees
Mathematical Subject Classification 2010
Primary: 20C30
Secondary: 20C15, 20C33
Received: 19 May 2014
Revised: 13 January 2015
Accepted: 23 February 2015
Published: 17 April 2015
Christine Bessenrodt
Institut für Algebra, Zahlentheorie und Diskrete Mathematik
Fakultät für Mathematik und Physik
Leibniz Universität Hannover
Welfengarten 1
D-30167 Hannover
Hung Ngoc Nguyen
Department of Mathematics
The University of Akron
Akron, OH 44325
United States
Jørn B. Olsson
Department of Mathematical Sciences
University of Copenhagen
DK-2100 Copenhagen Ø
Hung P. Tong-Viet
Department of Mathematics and Applied Mathematics
University of Pretoria
Private Bag X20
Hatfield, Pretoria 0002
South Africa