QUASIRECOGNITION BY PRIME GRAPH OF SOME ORTHOGONAL GROUPS OVER THE BINARY FIELD

Abstract

Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, p′ are joined by an edge if G has an element of order pp′. Let D = Dn(2), where n ≥ 3 or 2Dn(2), where n ≥ 15. In this paper we prove that D is quasirecognizable by prime graph, i.e. every finite group G with Γ(G) = Γ(D), has a unique nonabelian composition factor which is isomorphic to D. Finally, we consider the quasirecognition by spectrum for these groups. Specially we prove that if p = 2n + 1 ≥ 17 is a prime number, then Dp(2) is recognizable by spectrum.