Siberian Mathematical Journal

, Volume 54, Issue 3, pp 487–500

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

• Z. Momen
• B. Khosravi
Article

## 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 GB 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.

## Keywords

prime graph simple group recognition quasirecognition

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