Zeros of real irreducible characters of finite groups

Marinelli, Selena; Tiep, Pham

doi:10.2140/ant.2013.7.567

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Zeros of real irreducible characters of finite groups

Selena Marinelli and Pham Huu Tiep

Vol. 7 (2013), No. 3, 567–593

DOI: 10.2140/ant.2013.7.567

Abstract

We prove that if all real-valued irreducible characters of a finite group $G$ with Frobenius–Schur indicator $1$ are nonzero at all $2$ -elements of $G$ , then $G$ has a normal Sylow $2$ -subgroup. This result generalizes the celebrated Ito–Michler theorem (for the prime $2$ and real, absolutely irreducible, representations), as well as several recent results on nonvanishing elements of finite groups.

Keywords

real irreducible character, nonvanishing element, Frobenius–Schur indicator

Mathematical Subject Classification 2010

Primary: 20C15

Secondary: 20C33

Milestones

Received: 21 March 2011

Revised: 2 February 2012

Accepted: 16 March 2012

Published: 23 August 2013

Authors

Selena Marinelli
Dipartimento di Matematica L. Tonelli Università di Pisa Largo Bruno Pontecorvo 5 I-56127 Pisa Italy

Pham Huu Tiep
Department of Mathematics University of Arizona 617 N. Santa Rita Ave. P.O. Box 210089 Tucson AZ 85721-0089 United States
http://math.arizona.edu/~tiep/

Vol. 7, No. 3, 2013