The neutrosophic automorphisms of a neutrosophic groups  G (I) , denoted by Aut(G (I)) is a neu-trosophic group under the usual mapping composition. It is a permutation of  G (I) which is also a neutrosophic homomorphism. Moreover, suppose that X1 = X(G (I)) is the neutrosophic group of inner neutrosophic auto-morphisms of a neutrosophic group G (I) and Xn the neutrosophic group of inner neutrosophic automorphisms of Xn-1. In this paper, we show that if any neutrosophic group of the sequence G (I), X1, X2, … is the identity, then G (I) is nilpotent.

Let G = (V,E) be a simple graph with exactly n vertices and m edges. The aim of this paper is a new method for investigating nontriviality of the automorphism group of graphs. To do this, we prove that if |E| ³ ë (n - 1)2/2û then |Aut(G)| > 1 and |Aut(G)| is even number.

For a  nite group H, let cs(H) denote the set of non-trivial conjugacy class sizes of H and OC(H) be the set of the order components of H. In this paper, we show that if S is a  nite simple group with the disconnected prime graph and G is a  nite group such that cs(S) = cs(G), then jSj = jG=Z(G)j and OC(S) = OC(G=Z(G)). In particular, we show that for some  nite simple group S, G  = S  Z(G).

Let G be a group. For an element a 2 G, denote by C (a) the second centralizer of a in G, which is the set of all elements b 2 G such that bx = xb for every x 2 G that commutes with a. Let M be any maximal abelian subgroup of G. Then C 2 2 (a) ,M for every a 2 M. The abelian rank (a-rank) of M is the minimum cardinality of a set A ,M such that ∪,2 C a 2 A (a) generates M. Denote by Sn the symmetric group of permutations on the set X = f 1, : : :, n g. The aim of this paper is to determine the maximal abelian subgroups of Sn of a-rank 1 and describe a class of maximal abelian subgroups of Sn of a-rank at most 2.