Recognition of the simple groups $PSL_2(q)$ by character degree graph and order

Abstract

‎‎Let $G$ be a finite group‎, ‎and $Irr(G)$ be the set of complex irreducible characters of $G$‎. ‎Let $rho(G)$ be the set of prime divisors of character degrees of $G$‎. ‎The character degree graph of $G$‎, ‎which is denoted by $Delta(G)$‎, ‎is a simple graph with vertex set $rho(G)$‎, ‎and we join two vertices $r$ and $s$ by an edge if there exists a character degree of $G$ divisible by $rs$‎. ‎In this paper‎, ‎we prove that if $G$ is a finite group such that $Delta(G)=Delta(PSL_2(q))$ and $|G|=|PSL_2(q)|$‎, ‎then $GcongPSL_2(q)$‎.

Keywords

• ‎character degree graph‎
• ‎simple group‎
• ‎characterization