Zeros of $S$-characters

TL;DR

This paper investigates zeros of $S$-characters in finite groups, providing examples that challenge previous assumptions about their vanishing properties, and answers a question posed by Serre.

Contribution

It demonstrates that non-trivial $S$-characters can avoid vanishing on prime power order elements, contrary to transitive permutation characters.

Findings

Non-trivial $S$-characters do not necessarily vanish on prime power order elements.

Examples are provided that contradict the generalization from transitive permutation characters.

Answers a question posed by J-P. Serre regarding $S$-characters.

Abstract

The concept of $S$-characters of finite groups was introduced by Zhmud' as a generalisation of transitive permutation characters. Any non-trivial $S$-character takes a zero value on some group element. By a deep result depending on the classification of finite simple groups a non-trivial transitive permutation character even vanishes on some element of prime power order. We present examples that this does not generalise to $S$-characters, thereby answering a question posed by J-P. Serre.