Finite groups with the same Power graph
Group Theory
2020-10-29 v1
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 one is the power of the other. 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 them is nilpotent or has a normal Hall subgroup, the same is true with the other one.
Cite
@article{arxiv.2010.15038,
title = {Finite groups with the same Power graph},
author = {M. Mirzargar and R. Scapellato},
journal= {arXiv preprint arXiv:2010.15038},
year = {2020}
}
Comments
9 pages