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Groups with the same prime graph as the orthogonal group B n (3)

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Abstract

Let G be a finite group. The prime graph of G is denoted by Γ(G). It is proved in [1] that if G is a finite group such that Γ(G) = Γ(B p (3)), where p > 3 is an odd prime, then G ⋟ B p (3) or C p (3). In this paper we prove the main result that if G is a finite group such that Γ(G) = Γ(B n (3)), where n ≥ 6, then G has a unique nonabelian composition factor isomorphic to B n (3) or C n (3). Also if Γ(G) = Γ(B 4(3)), then G has a unique nonabelian composition factor isomorphic to B 4(3), C 4(3), or 2 D 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.

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Authors and Affiliations

  1. Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran

    Z. Momen & B. Khosravi

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  1. Z. Momen
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  2. B. Khosravi
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Correspondence to Z. Momen.

Additional information

Original Russian Text Copyright © 2013 Momen Z. and Khosravi B.

The second author was supported in part by the IPM (the Institute for Research in Fundamental Sciences, Iran) (Grant 91050116).

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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 54, No. 3, pp. 620–636, May–June, 2013.

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Momen, Z., Khosravi, B. Groups with the same prime graph as the orthogonal group B n (3). Sib Math J 54, 487–500 (2013). https://doi.org/10.1134/S0037446613030142

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