On Quasi Steinberg characters of Symmetric and Alternating groups and their Double Covers

TL;DR

This paper classifies quasi p-Steinberg characters of symmetric and alternating groups, including their double covers, revealing bounds on their existence and character properties.

Contribution

It provides a complete classification of quasi p-Steinberg characters for symmetric, alternating groups, and their double covers, with explicit bounds on their sizes.

Findings

Non-linear quasi p-Steinberg characters exist only for small n (up to 8 for S_n, 9 for A_n).

The classification includes the double covers of these groups.

The existence of such characters is tightly constrained by the group size.

Abstract

An irreducible character of a finite group $G$ is called quasi $p$-Steinberg character for a prime $p$ if it takes a nonzero value on every $p$-regular element of $G$. In this article, we classify the quasi $p$-Steinberg characters of Symmetric ($S_n$) and Alternating ($A_n$) groups and their double covers. In particular, an existence of a non-linear quasi $p$-Steinberg character of $S_n$ implies $n \leq 8$ and of $A_n$ implies $n \leq 9$.