On quasi Steinberg characters of symmetric and alternating groups and their double covers

Abstract

An irreducible character of a finite group [Formula: see text] is called quasi [Formula: see text]-Steinberg character for a prime [Formula: see text] if it takes a nonzero value on every [Formula: see text]-regular element of [Formula: see text]. In this paper, we classify the quasi [Formula: see text]-Steinberg characters of Symmetric ([Formula: see text]) and Alternating ([Formula: see text]) groups and their double covers. In particular, an existence of a nonlinear quasi [Formula: see text]-Steinberg character of [Formula: see text] implies [Formula: see text] and of [Formula: see text] implies [Formula: see text].