Real Constituents of Permutation Characters

TL;DR

This paper generalizes Burnside's theorem on real characters using permutation characters, provides new characterizations of 2-closed groups, and classifies certain primitive permutation groups with fixed points for real elements.

Contribution

It extends classical results on real characters, introduces a new characterization of 2-closed groups, and classifies primitive permutation groups with fixed points for real elements.

Findings

Generalization of Burnside's theorem on real characters

New characterization of 2-closed finite groups

Classification of primitive permutation groups with fixed points for real elements

Abstract

We prove a broad generalization of a theorem of W. Burnside on real characters using permutation characters. Under a necessary hypothesis, We can give some control on multiplicities (a result that needs the Classification of Finite Simple Groups). Along the way, we also give a new characterization of the 2-closed finite groups using odd-order real elements of the group. All this can be seen as a contribution to Brauer's Problem 11. We also obtain similar results for 2-Brauer characters. We also classify finite primitive permutation groups in which every real element has a fixed point.