In the vast domain of group theory, the study of automorphisms stands as a cornerstone in understanding the symmetries and structural properties of algebraic systems. In this research, we delve deep into the intricate properties of the automorphism group of a given graph or structure G, with a primary focus on understanding the measure d(G), defined as the difference between the order of the automorphism group and the graph itself: d(G) =  ∣ Aut(G) ∣  −  ∣ G∣. Beginning with a pivotal examination, we scrutinize the isomorphism between the automorphism group of the symmetric group Sn and the symmetric group itself for n > 6. This foundational problem serves as a springboard to more generalized scenarios where d(G) takes values such as ±1, ±p, and ±2p. Through the innovative lens of the “d(G)” measure, we unearth novel properties of automorphism groups that were previously elusive. This study not only enhances the mathematical understanding of automorphisms but also presents a fresh perspective on related algebraic structures.