This paper discusses how to express a finite group as graph, specifically about the identity graph of cyclic group. The term chosen for is an because it element that holds key in forming graph. Through can be seen which elements are inverse themselves and other properties We will look characteristics group, both cases odd even order.

In this paper we define S algebras and show that every finite group can be found in some S algebra.  We define and study the S degree of a finite group and determine the S degree of several classes of finite groups such as cyclic groups, elementary abelian $p$-groups, and dihedral groups $D_p$.

The article introduces cyclic dilation groups and finite affine groups for prime integers, and  as an application of this theory it presents a unified group theoretical approach for the  cyclic wavelet transform (CWT) of prime dimensional periodic signals.