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

چکیده مقاله

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 {equal to or succeeds} 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. © 2013 Pleiades Publishing, Ltd.