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Sibirskii Matematicheskii Zhurnal, 2007, Volume 48, Number 3, Pages 707–716
(Mi smj59)
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This article is cited in 35 scientific papers (total in 35 papers)
Quasirecognition by prime graph of the simple group $^2G_2(q)$
A. Khosravia, B. Khosravibc a 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.
Received: 27.10.2005 Revised: 09.02.2006
Citation:
A. Khosravi, B. Khosravi, “Quasirecognition by prime graph of the simple group $^2G_2(q)$”, Sibirsk. Mat. Zh., 48:3 (2007), 707–716; Siberian Math. J., 48:3 (2007), 570–577
Linking options:
https://www.mathnet.ru/eng/smj59 https://www.mathnet.ru/eng/smj/v48/i3/p707
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