The Independent Number of the Co-prime Power Order Graph of finite Groups

In relation to a given group, its vertex set V(β_G) characterizes the co-prime power order graph of that group, denoted by G and represented by β_G. If and only if gcd (|x|,|y|) = p^m, where p is a prime number and m is a non-negative integer, then the vertices x and y are adjacent. This paper established to work on the graph β_G, and its induced graphs to determine the independence set and the independence number α(β_G). All of these notions focus on the order of vertex set V(β_G)  of a finite group G, and the independent set, and α(β_G) are depend on some properties of  the graph βG as the fullness of the generalized quaternion group, dihedral group, and cyclic group.