Perfect codes in proper intersection power graphs of finite groups

Given a finite group G with the identity e, the proper intersection power graph of G is the graph with vertex set \(G\setminus \{e\}\), in which two distinct vertices x and y are adjacent if \(\langle x\rangle \cap \langle y\rangle \) is non-trivial. In this paper, we give a necessary and sufficient condition for a proper intersection power graph to contain a perfect code. As applications, we classify all finite nilpotent groups whose proper intersection power graphs admit a perfect code. We also classify a few classes of finite non-nilpotent groups whose proper intersection power graphs admit a perfect code, such as, symmetric groups and alternating groups.