ON RECOGNITION BY PRIME GRAPH OF THE PROJECTIVE SPECIAL LINEAR GROUP OVER GF(3)

Bahman Khosravi, Behnam Khosravi, Hamid Reza Dalili Oskouei

Abstract: Let $G$ be a finite group. The prime graph of $G$ is denoted by $\Gamma(G)$. We prove that the simple group $\PSL_n(3)$, where $n\geq 9$, is quasirecognizable by prime graph; i.e., if $G$ is a finite group such that $\Gamma(G)=\Gamma(\PSL_n(3))$, then $G$ has a unique nonabelian composition factor isomorphic to $\PSL_n(3)$. Darafsheh proved in 2010 that if $p>3$ is a prime number, then the projective special linear group $\PSL_p(3)$ is at most 2-recognizable by spectrum. As a consequence of our result we prove that if $n\geq 9$, then $\PSL_n(3)$ is at most $2$-recognizable by spectrum.

Keywords: prime graph, simple group, recognition, quasirecognition

Classification (MSC2000): 20D05, 20D60; 20D08

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