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 paper, 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 nonlinear quasi $p$-Steinberg character of $S_n$ implies $n\le 8$ and of $A_n$ implies $n\le 9$.

中文翻译：

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对称交错群的拟Steinberg特征及其双重覆盖

有限群的不可约特征$G$被称为准$p$- 素数的斯坦伯格字符$p$如果它对每个都取非零值$p$- 的常规元素$G$. 在本文中，我们将准$p$-Steinberg 的对称字符（${\mathrm{小号}}_n$) 和交替 (${\mathrm{一个}}_n$) 组及其双重封面。特别是，非线性准的存在$p$——斯坦伯格的性格${\mathrm{小号}}_n$暗示$n\le 8$和${\mathrm{一个}}_n$暗示$n\le 9$.