On Some Applications and Open Problems about (m-Groups)

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.