Abstract:The power graph P (G) of a group G is a graph with vertex set G, where two vertices u and v are adjacent if and only if u = v and u m = v or v m = u for some positive integer m. In this paper, we raise and study the following question: For which natural numbers n every two groups of order n with isomorphic power graphs are isomorphic? In particular, we determine prove that all such n are cube-free and are not multiples of 16. Moreover, we show that if two finite groups have isomorphic power graphs and one of t… Show more
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