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 Sibirsk. Mat. Zh., 2013, Volume 54, Number 3, Pages 620–636 (Mi smj2447)

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.

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English version:
Siberian Mathematical Journal, 2013, 54:3, 487–500

Bibliographic databases:

UDC: 512.54

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

Citation in format AMSBIB
\Bibitem{MomKho13} \by Z.~Momen, B.~Khosravi \paper Groups with the same prime graph as the orthogonal group $B_n(3)$ \jour Sibirsk. Mat. Zh. \yr 2013 \vol 54 \issue 3 \pages 620--636 \mathnet{http://mi.mathnet.ru/smj2447} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3112620} \transl \jour Siberian Math. J. \yr 2013 \vol 54 \issue 3 \pages 487--500 \crossref{https://doi.org/10.1134/S0037446613030142} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000322243600014} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84881078485} 

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This publication is cited in the following articles:
1. Khosravi B., Khosravi B., Oskouei Hamid Reza Dalili, “on Recognition By Prime Graph of the Projective Special Linear Group Over Gf(3)”, Publ. Inst. Math.-Beograd, 95:109 (2014), 255–266
2. Momen Z., Khosravi B., “Quasirecognition By Prime Graph of the Simple Group Bn (9)”, Proc. Rom. Acad. Ser. A-Math. Phys., 16:3 (2015), 397–404
3. Mahmoudifar A., Khosravi B., “on Quasirecognition By Prime Graph of the Simple Groups a(N)(+) (P) and a(N)(-) (P)”, J. Algebra. Appl., 14:1 (2015), 1550006
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