On finite groups whose power graph is claw-free

2024/07/29 by Manna, Pallabi, Mandal, Santanu, Lucchini, Andrea
#FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2407.20110

Abstract

A graph is called claw-free if it contains no induced subgraph isomorphic to the complete bipartite graph K_{1, 3}. The undirected power graph of a group G has vertices the elements of G, with an edge between g₁ and g₂ if one of the two cyclic subgroups ⟨ g₁⟩, ⟨ g₂⟩ is contained in the other. It is denoted by P(G). The reduced power graph, denoted by P^*(G), is the subgraph of P(G) induced by the non-identity elements. The main purpose of this paper is to explore the finite groups whose reduced power graph is claw-free. In particular we prove that if P^*(G) is claw-free, then either G is solvable or G is an almost simple group. In the second case the socle of G is isomorphic to PSL(2,q) for suitable choices of q. Finally we prove that if P^*(G) is claw-free, then the order of G is divisible by at most 5 different primes.

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