On The Noncommuting Graph Of Groups

In this paper, we investigate some basic properties of the noncommuting graph associated with a finite group and their effects on the structure of the group.Let G be a finite group. We denote the noncommuting graph of G byâ–½(G) and in which the vertex set V(G):= G \ Z(G) and x, y∈V(G) form an edge if and only if [x, y]≠1, denoted by x～y.The number of edges incident with a vertex g in a graph is called the degree of g and denoted by deg(g). For the noncommuting graphâ–½(G), we put p(G) = (?) deg(g).Two graphsâ–½(G₁) andâ–½(G₂) are said to be isomorphic if there exists a one-to-one onto mapping:φ: V(G₁)→V(G₂), such that u～v (?)φ(u)～φ(v) for all u, v∈V(G₁).Such a mapφis called a graph isomorphism. For isomorphic graphsâ–½(G₁) andâ–½(G₂), we denote them byâ–½(G₁)≌▽(G₂).Forâ–½(G), we obtain the follow results:Theorem 3. 1 Let G be an inner-nilpotent group of order p^αq^β(p, q are distinct primes), and its Sylow q-subgroup Q is normal in G. Ifâ–½(H)≌▽(G) and |H| = |G|, then Sylow p-subgroup P₁ of H is abelian; furthermore, Sylow q-subgroup Q₁ of H is normal in H when p＜q, moreover, Q₁ is abelian if Q is abelian.Theorem 3. 2 Let p be a prime. Ifâ–½(G)≌▽(D_4p), then G≌D_4p or G≌Q_4p.Theorem 3. 3 (1)Ifâ–½(G)≌▽(A₄), then G≌A₄. (2) Let p be aprime and p≥5. Ifâ–½(G)≌▽(A_n), n =p, p+1, p+2, then G≌A_n.Theorem 4. 1 Let G be a nonabelian finite group and |G| = 2^an with n odd. Let P∈Syl₂(G). If P has exactly one involution, thenâ–½(G) is Euler if and only if there exists an involution in Z(G).