The Power Graph of a Torsion-Free Group Determines the Directed Power\n Graph

2020/06/02 by Samir Zahirović, Zahirović, Samir
Computer Science · Mathematics · #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2006.01984

openalex publication_date 2020/06/02 · openalex created_date 2022/07/26 · openalex updated_date 2026/08/01

Abstract

The directed power graph \ mathcal G( mathbf G) of a group mathbf G\nis the simple digraph with vertex set G such that x\→ y if y is\na power of x. The power graph of mathbf G, denoted with mathcal\nG( mathbf G), is the underlying simple graph.\n In this paper, for groups mathbf G and mathbf H, the following is\nproved. If mathbf G has no quasicyclic subgroup mathbf C_p^\∞ which\nhas trivial intersection with every cyclic subgroup mathbf K of mathbf G\nsuch that mathbf K not\≤ mathbf C_p^\∞, then mathcal G( mathbf\nG)\≅ mathcal G( mathbf H) implies \ mathcal G( mathbf G)\≅\n\ mathcal G( mathbf H). Consequently, any two torsion-free groups having\nisomorphic power graphs have isomorphic directed power graphs.\n

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