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Sibirskii Matematicheskii Zhurnal, 2013, Volume 54, Number 3, Pages 620–636
(Mi smj2447)
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This article is cited in 4 scientific papers (total in 4 papers)
Groups with the same prime graph as the orthogonal group $B_n(3)$
Z. Momen, B. Khosravi Dept. of Pure Math., Faculty of Math. and Computer Sci., Amirkabir University of Technology (Tehran Polytechnic), 424, Hafez Ave., Tehran 15914, Iran
Abstract:
Let $G$ be a finite group. The prime graph of $G$ is denoted by $\Gamma(G)$. It is proved in [1] that if $G$ is a finite group such that $\Gamma(G)=\Gamma(B_p(3))$, where $p>3$ is an odd prime, then $G\ge B_p(3)$ or $C_p(3)$. In this paper we prove the main result that if $G$ is a finite group such that $\Gamma(G)=\Gamma(B_n(3))$, where $n\ge6$, then $G$ has a unique nonabelian composition factor isomorphic to $B_n(3)$ or $C_n(3)$. Also if $\Gamma(G)=\Gamma(B_4(3))$, then $G$ has a unique nonabelian composition factor isomorphic to $B_4(3)$, $C_4(3)$, or $^2D_4(3)$. It is proved in [2] that if $p$ is an odd prime, then $B_p(3)$ is recognizable by element orders. We give a corollary of our result, generalize the result of [2], and prove that $B_{2k+1}(3)$ is recognizable by the set of element orders. Also the quasirecognition of $B_{2k}(3)$ by the set of element orders is obtained.
Keywords:
prime graph, simple group, recognition, quasirecognition.
Received: 25.08.2011
Citation:
Z. Momen, B. Khosravi, “Groups with the same prime graph as the orthogonal group $B_n(3)$”, Sibirsk. Mat. Zh., 54:3 (2013), 620–636; Siberian Math. J., 54:3 (2013), 487–500
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https://www.mathnet.ru/eng/smj2447 https://www.mathnet.ru/eng/smj/v54/i3/p620
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Abstract page: | 271 | Full-text PDF : | 76 | References: | 52 | First page: | 2 |
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