Shahid Rajaee Teacher Training University Journal of Discrete Mathematics and Its Applications 2981-0809 11 2 2026 06 01 Finite groups whose enhanced power graphs are unique 153 161 12568 10.22061/jdma.2026.12605.1171 EN Mahsa Mirzargar Faculty of Science, Mahallat Institute of Higher Education, Mahallat, I. R. Iran Journal Article 2025 10 10 The enhanced power graph Pe(G) of a group G is the graph whose vertex set is G, with two elements u and v adjacent if there is an element z ∈ G such that ⟨u, v⟩ = ⟨z⟩. In this paper, we investigate classes of groups whose enhanced power graphs uniquely determine their structure; that is, if Pe(G) ≅ Pe(H), then G ≅ H. We also study the set of natural numbers n for which every group of order n is uniquely determined (up to isomorphism) by its enhanced power graph. We consider groups that have the same number of elements of each order and exploit necessary conditions to identify situations in which a property of a group G is preserved by all groups sharing the same enhanced power graph. In particular, we show that if two finite groups have isomorphic enhanced power graphs and one of them is nilpotent or has a normal Hall subgroup, then the other must also share that property.

https://jdma.sru.ac.ir/article_12568_8c665163ee23ba68c10117a4f50d2d2e.pdf