A new characterization of $L_{3}(p)$

Abstract

Given a finite group $G$, let $V(G)=\{p^{e_{p}(G)}|p\in\rho(G)\}$, where $\rho(G)$ is the set of prime divisors of the degrees of all irreducible characters of $G$ and $p^{e_{p}(G)}=\mathrm{max}\{\chi(1)_{p}|\chi\in \mathrm{Irr}(G)\}$. In fact, $V(G)$ is the vertex set of the prime-power graph of $G$. An interesting topic is to study if a finite simple group $M$ can be uniquely determined by its order $|M|$ and $V(M)$. It has been proved that the simple groups $L_{2}(p^{2})$ and $L_{2}(p^{3})$ can be uniquely determined by its orders and vertex set of its prime-power graphs, respectively, where $p$ is a prime. In this paper, we continue this topic and show that $G\cong L_{3}(p)$ if and only if $|G|=|L_{3}(p)|$ and $V(G)=V(L_{3}(p))$, where $p$ is a prime.

Keywords

• simple group
• irreducible character
• degree

Supporting Institution

This research was supported by Guangxi Natural Science Foundation (2022GXNSFBA035572), National Natural Science Foundation of China (12301021,12401018), Chongqing Technology and Business University (2353005), Doctoral Through Train Scientific Research Project of Chongqing (sl202100000324).

Ethical Statement

This paper is new. Neither the entire paper nor any part of its content has been published or has been accepted elsewhere. It is not being submitted to any other journal. All authors have seen the manuscript and approved to submit to your journal.

Thanks

Thank you very much for your attention and consideration, and please direct all correspondence about this manuscript to me or Yu Li ([e-mail]).

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