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

Document Type: Research Paper

Authors

1 Faculty of Mathematics and Computer science, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran

2 Faculty of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), 15914 Tehran, Iran

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 $G\cong\PSL_2(q)$‎.

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