Quasirecognition by Prime Graph of the Groups 2 D 2 n ( q ) Where q < 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 Dn(q), where n 6= 4k, are quasirecognizable by prime graph. Now in this paper we discuss the quasirecognizability by prime graph of the simple groups D2k(q), where k ≥ 9 and q is a prime power less than 105.


Introduction
Let G be a finite group.By π e (G) we denote the set of element orders of G.For an integer n we define π(n) as the set of prime divisors of n and we set π(G) for π(|G|).The prime graph of the Gruenberg-Kegel graph of G is denoted by Γ(G) and it is a graph with vertex set π(G) in which two distinct vertices p and q are adjacent if and only if pq ∈ π(G), and in this case we will write p ∼ q.
A subset of vertices of Γ(G) is called an independent subset of Γ(G) if its vertices are pairwise nonadjacent.Denote by t(G) the maximal number of primes in π(G) pairwise nonadjacent in Γ(G).We also denote by t(2, G) the maximal number of vertices in the independent sets of Γ(G) containing 2. A finite nonabelian simple group P is called quasirecognizable by prime graph if every finite group G with Γ(G) = Γ(P) has a composition factor isomorphic to P. P is called recognizable by prime graph if Γ(G) = Γ(P) implies G ∼ = P.In addition, finite group P is considered to be recognizable by a set of element orders if the equality π e (G) = π e (P), for each finite group G implies that G ∼ = P.A finite simple nonabelian group P is considered to be quasirecognizable by the set of element orders if each finite group H with π e (H) = π e (P) has a composition factor isomorphic to P. If a finite simple group is quasirecognizable (recognizable) by prime graph, then it is quasirecognizable (recognizable) by set of element orders, but the inverse is not true necessarily and proving by prime graph is more difficult.
Hagie determined finite groups H satisfying Γ(H) = Γ(S), where S is a sporadic simple group [1].In [2,3], finite groups with the same prime graph as Γ(PSL(2, q)), where q is a prime power, are determined.Quasirecognizability by prime graph of groups G 2 (3 2n+1 ) and 2 B 2 (2 2n+1 ) has been proved in [4].In [5][6][7], finite groups with the same prime graphs as ) and Γ( 2 D 2k (3)) are obtained.In addition, in [8], it is proved that if p is a prime less than 1000, for suitable n, the finite simple groups L n (p) and U n (p) are quasirecognizable by prime graph.Now as the main result of this paper, we prove the following theorem: Main Theorem.The finite simple group 2 D 2k (q), where k ≥ 9 and q < 10 5 , is quasirecognizable by prime graph.
Throughout this paper, all groups are finite and by a simple group we mean a nonabelian simple group.All further unexplained notations are standard and the reader is referred to [9].

Preliminary Results
Lemma 1 ([10] Theorem 1).Let G be a finite group with t(G) ≥ 3 and t(2, G) ≥ 2. Then the following hold: 1. there exists a finite nonabelian simple group S such that for the maximal normal soluble subgroup K of G. 2. for every independent subset ρ of π(G) with |ρ| ≥ 3 at most one prime in ρ divides the product |K| • | Ḡ/S|.
In particular, t(S) ≥ t(G) − 1. 3. one of the following holds: and S ∼ = A 7 or L 2 (q) for some odd q.
Remark 1.In Lemma 1, for every odd prime p ∈ π(S), we have t(p, S) If q is a natural number, r is an odd prime and (q, r) = 1, then by e(r, q) we denote the smallest natural number m such that q m ≡ 1 (mod r).Given an odd q, put e(2, q) = 1 if q ≡ 1 (mod 4) and put e(2, q) = 2 if q ≡ −1 (mod 4).Using Fermat's little theorem we can see that if r is an odd prime such that r | (q n − 1), then e(r, q) | n.
, where q is power of prime p. Define Suppose r, s are odd primes and r, s ∈ π(D ε n (q)) \ {p}.Put k = e(r, q), l = e(s, q), and 1 ≤ η(k) ≤ η(l).Then r and s are non-adjacent if and only if 2η(k) + 2η(l) > 2n − (1 − ε(−1) k+l ) and k and l satisfy the following condition: l k is not an odd integer, and if ε = +, then the chain of equalities is not true.

Lemma 3 ([11] Proposition 2.3).
Let G be one of the simple groups of Lie type, B n (q) or C n (q), over a field of characteristic p. Define Let r, s be odd primes with r, s ∈ π(G) \ {p}.Put k = e(r, q) and l = e(s, q), and suppose that 1 ≤ η(k) ≤ η(l).Then r and s are non-adjacent if and only if η(k) + η(l) > n, and k, l satisfy: l k is not an odd natural number.Lemma 4 ([12] Proposition 2.1).Let G = L n (q), where q is a power of prime p.Let r and s be odd primes and r, s ∈ π(G) \ {p}.Put k = e(r, q) and l = e(s, q) and assume that 2 ≤ k ≤ l.Then r and s are nonadjacent if and only if k + l > n and k does not divide l.Lemma 5 ([12] Proposition 2.2).Let G = U n (q), where q is a power of prime p. Define Let r and s be odd primes and r, s ∈ π(G) \ {p}.Put k = e(r, q) and l = e(s, q) and suppose that 2 ≤ ν(k) ≤ ν(l).Then r and s are nonadjacent if and only if ν(k) + ν(l) > n and ν(k) does not divide ν(l).
For Lemmas 2 and 5, simultaneously, we define the following function: which we will use in the proofs.We note that a prime r with e(r, q) = m is called a primitive prime divisor of q m − 1 (obviously, q m − 1 can have more than one primitive prime divisors).Lemma 6. (Zsigmondy's theorem) [13] Let p be a prime and let n be a positive integer.Then one of the following holds: 1. there is a primitive prime p for p n − 1, that is , p | (p n − 1) but p | (p m − 1), for every 1 ≤ m < n, 2. p = 2, n = 1 or 6, 3. p is a Mersenne prime and n = 2.

Proof of the Main Theorem
Throughout this section, we suppose that D := 2 D 2k (p α ) where k ≥ 9, 2 < p α < 10 5 and G is a finite group such that Γ(G) = Γ(D).We denote a primitive prime divisor of q i − 1 by r i and a primitive prime divisor of q i − 1 by r i , where q = q.
By ([12] Tables 4, 6 and 8), we deduce that t(D) ≥ 14 and t(2, D) ≥ 2. Therefore, t(G) ≥ 14 and t(2, G) ≥ 2. Now by Lemma 1, it follows that there exists a finite nonabelian simple group S such that where K is the maximal normal solvable subgroup of G.In addition, t(S) ≥ t(G) − 1 and t(2, S) ≥ t(2, G) by Lemma 1.Therefore, t(S) ≥ 13 and t(2, S) ≥ 2. On the other hand, by ( [12] Tables 2 and 9), if S is isomorphic to a sporadic or an exceptional simple group of Lie type, then t(S) ≤ 12.This implies that S is not isomorphic to any sporadic or any exceptional simple groups of Lie type.
In the sequel, we consider each possibility for S.
Lemma 7. S is not isomorphic to any alternating group.
Lemma 8.If S is isomorphic to a classical simple group of Lie type over a field of characteristic p, then S ∼ = D.
Proof.Let S be a nonabelian simple group of Lie type over GF(q ), q = p β .By the hypothesis, where N is the maximal normal solvable subgroup of G.In the sequel, we denote by r i a primitive prime divisor of q i − 1 and by r i a primitive prime divisor of q i − 1.We remark that {p, r 2n } ⊆ π(S) and |ρ(p, G) ∩ π(S)| ≥ 3 by Lemma 1.Now we consider the following cases: Case 1.Let r 2n−2 ∈ π(S).In addition, let p 1 and p 2 be two primitive prime divisors of p (2n−2)α − 1 and p 2nα − 1, respectively.So we may assume that p 1 and p 2 are r 2n−2 and r 2n , respectively.This implies that {r 2n−2 , r 2n } ⊆ π(S).Thus r 2n−2 is a primitive prime divisor of q s − 1 and r 2n is a primitive prime divisor of q t − 1, where s = e(r 2n−2 , p β ) and t = e(r 2n , p β ).It follows that (2n − 2)α | sβ and 2nα | tβ.On the other hand, using Zsigmondy's theorem, we conclude that tβ ≤ 2nα and so tβ = 2nα.Furthermore, since 2n < 2(2n − 2), we have sβ = (2n − 2)α and s < t.Now we consider each possibility for S, separately.If ρ(p, S) = {r i | i ∈ I} ∪ {p}, then using the results in [12], each r j ∈ π(S), where j ∈ I is adjacent to p in Γ(S).
Similarly, we can prove that S ∼ = 2 D m (q ), where m is even and S ∼ = D m (q ).