Rational representations and permutation representations of finite groups

TL;DR

This paper explores which rational-valued characters of finite groups can be expressed as integer linear combinations of permutation characters, providing solutions for quasi-elementary groups and applications to simple groups.

Contribution

It offers a solution to the problem for quasi-elementary groups and demonstrates that certain simple groups can have arbitrarily large minimal multiples for permutation representations.

Findings

Characterization of Q-valued characters as Z-linear combinations of permutation characters.

Solution provided for quasi-elementary groups.

Existence of simple groups with arbitrarily large minimal multiples for permutation representations.

Abstract

We investigate the question which Q-valued characters and characters of Q-representations of finite groups are Z-linear combinations of permutation characters. This question is known to reduce to that for quasi-elementary groups, and we give a solution in that case. As one of the applications, we exhibit a family of simple groups with rational representations whose smallest multiple that is a permutation representation can be arbitrarily large.