Finite groups whose enhanced power graphs are unique

Mirzargar, Mahsa

doi:10.22061/jdma.2026.12605.1171

Finite groups whose enhanced power graphs are unique

Document Type : Full Length Article

Author

mahsa mirzargar

Mahallat Institute of higher Education

10.22061/jdma.2026.12605.1171

Abstract

The enhanced power graph $ P_e(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 $z\in G$ such that $ \langle u, v \rangle = \langle z\rangle $.
In this paper, we investigate classes of groups whose enhanced power graphs uniquely determine their structure; that is, if $ P_e(G) \cong P_e(H) $, then $ G \cong 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.

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