Galoitica: Journal of Mathematical Structures and Applications

The generalizations of abelian groups have been studied widely because of their importance in classification theorem and representation. A group G is called an m-power closed group or (m-group) if and only if it has the following property xm ym=zm  ∀x,y ∈ G and for z ∈ G. This paper studies a special case of m-groups, when G is a finite m-group and n-group at the same time with relatively prime integers m and n, which is called a Monic group. It presents the necessary and sufficient conditions for a monic group G to be cyclic, abelian, nilpotent, and solvable by the corresponding property of its power subgroups Gm , Gn. Also, this work introduces three open problems in the theory of finite groups.

The aim of this paper is to solve the “ON MY TURF" game over some finite nonabelian groups. Also, it presents the following results: 1-) If G has odd order, and the set F contains the identity element, then the first player A has a winning strategy. If F does not contain the identity, then B has a winning strategy. 2-) If G has an even order with only one element of order two, there is a winning strategy related to set F. 3-) If G has an even order with only three elements of order two which generate a subgroup isomorphic to Z2 × Z2, there is a winning strategy related to the set F.

This paper uses the algebraic structure of the group to introduce a novel algebraic game with three players that occurs over finite groups.  Also, it analyzes this game over some finite groups with orders up to ten.

The objective of this paper is to present a new class of Diophantine equations derived from the group of units problem of n-cyclic refined neutrosophic rings of integers by using homomorphisms between these rings and a finite Cartesian product ring of Z with itself. Also, this work provides many examples about this class and its solvability as a new application of neutrosophic algebraic structures in number theory.

 This paper is devoted to defining a new generalization of m-power closed groups by using some special mappings defined on a group (power maps). Also, a new group will be presented according to these mappings which will be related to other algebraic structures.