The Cubic Dihedral Permutation Groups of Order 4k

Abstract

Given any mathematical system with identity and any positive integer k≥1, the structure of any dihedral group is generated by only two elements which are satisfy some conditions and belong to the set of mathematical system. Here in this work, we will give new structure to consider new groups are called cubic dihedral groups which are generated by three elements <a,b,c>instead of two elements <a,b>. That means we make to solve all the problems that maybe appear when we increase the elements to generate this new structure of groups. Next, new class of cubic dihedral groups is given they are called cubic dihedral permutation groups. Also, in this new class the specifying condition of positive integer k≥1 to get cubic dihedral permutation groups is shown as k=ord(a). This condition is important and we explained that the structure is not necessary satisfy requirements for group if k≠ord(a).