This article is cited in 35 papers

Quasirecognition by prime graph of the simple group $^2G_2(q)$

A. Khosravi^a, B. Khosravi<sup>bc

a</sup> University for Teacher Education
^b Institute for Studies in Theoretical Physics and Mathematics
^c Dept. of Pure Math., Faculty of Math. and Computer Sci., Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran

Abstract: Let $G$ be a finite group. The main result of this paper is as follows: If $G$ is a finite group, such that $\Gamma(G)=\Gamma(^2G_2(q))$, where $q=3^{2n+1}$ for some $n\ge 1$, then $G$ has a (unique) nonabelian composition factor isomorphic to $^2G_2(q)$. We infer that if $G$ is a finite group satisfying $|G|=|^2G_2(q)|$ and $\Gamma(G)=\Gamma(^2G_2(q))$ then $G\cong{}^2G_2(q)$. This enables us to give new proofs for some theorems; e.g., a conjecture of W. Shi and J. Bi. Some applications of this result are also considered to the problem of recognition by element orders of finite groups.

Keywords: quasirecognition, prime graph, simple group, element orders.

UDC: 519.542

Received: 27.10.2005
Revised: 09.02.2006

 English version: Siberian Mathematical Journal, 2007, 48:3, 570–577

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