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 L = Ln(2) or Un (2), where n 17. We prove that L is quasirecognizable by prime graph, i.e. if G is a finite group such that Γ(G) = Γ(L), then G has a unique nonabelian composition factor isomorphic to L. As a consequence of our result we give a new proof for the recognition by element orders of Ln(2). Also we conclude that the simple group Un(2) is quasirecognizable by element orders.

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.

Abstract. Let G be a finite group. The prime graph Γ(G) of G is defined as follows: The set of vertices of Γ(G) is the set of prime divisors of |G| and two distinct vertices p and p are connected in Γ(G), whenever G has an element of order pp . A non-abelian simple group P is called recognizable by prime graph if for any finite group G with Γ(G) = Γ(P ), G has a composition factor isomorphic to P . In [4] proved finite simple groups 2 D n (q), where n = 4k are quasirecognizable by prime graph. Now in this paper we discuss the quasirecognizability by prime graph of the simple groups 2 D 2k (q), where k ≥ 9 and q is a prime power less than 10 5 .

Let G be a finite group. The prime graph Γ(G) of G is defined as follows: The set of vertices of Γ(G) is the set of prime divisors of |G| and two distinct vertices p and p are connected in Γ(G), whenever G contains an element of order pp. A non-abelian simple group P is called recognizable by prime graph if for any finite group G with Γ(G) = Γ(P), G has a composition factor isomorphic to P. It is been proved that finite simple groups 2 D n (q), where n = 4k, are quasirecognizable by prime graph. Now in this paper we discuss the quasirecognizability by prime graph of the simple groups 2 D 2k (q), where k ≥ 9 and q is a prime power less than 10 5 .