In this paper, we introduce some properties of an intuitionistic normal fuzzy m-subgroup of m- group with m-homomorphism and isomorphism. We study the image, the pre-image and the inverse mapping of the intuitionistic normal fuzzy m-subgroups.

The aim of this paper is the study of applications of a group in a general fuzzy automaton. For this purpose, we de ne the concepts of fuzzy normal kernel of a general fuzzy automaton, fuzzy kernel of a general fuzzy automaton, adjustable, multiplicative. Then we obtain the relationships between them...

In this paper, we define the term " η-fuzzy subgroup" and show that every fuzzy subgroup is a η-fuzzy subgroup. We define some of the algebraic properties of the concept of η-fuzzy cosets. Furthermore, we initiate the study of the η-fuzzy normal subgroup and the quotient group with respect  to the η-fuzzy normal subgroup and demonstrate some of their various group theoretical properties.

Let G be a finite group and H, K be two subgroups of G. We introduce the relative non-normal graph of K with respect to H, denoted by NH, K, which is a bipartite graph with vertex sets H\HK and K\NK(H) and two vertices x ∈ H\HK and y ∈ K\NK(H) are adjacent if xy /∈ H, where HK = ∩ k∈ K Hk and NK(H) = {k ∈ K: Hk = H}. We determined some numerical invariants and state that when this graph is planar or outerplanar.

The number of subgroups, normal subgroups and characteristic subgroups of a finite group G are denoted by Sub(G), NSub(G) and CSub(G), respectively. The main goal of this paper is to present a matrix model for computing these positive integers for dicyclic groups, semi-dihedral groups, and three sequences U6n, V8n and H(n) of groups that can be presented as follows: U6n = ⟨ a, b | a2n = b3 = e, bab = a⟩ , V8n = ⟨ a, b | a2n = b4 = e, aba = b− 1, ab− 1a = b⟩ , H(n) = ⟨ a, b, c | a2n− 2 = b2 = c2 = e, [a, b] = [b, c] = e, ac = ab⟩ . For each group, a matrix model containing all information is given.

We study soluble groups G in which each subnormal subgroup H with in, nite rank is commensurable with a normal subgroup, i. e. there exists a normal subgroup N such that H \ N has , nite index in both H and N. We show that if such a G is periodic, then all subnormal subgroups are commensurable with a normal subgroup, provided either the Hirsch-Plotkin radical of G has in, nite rank or G is nilpotent-by-abelian (and has in, nite rank).