On finite groups whose power graph is claw-free

Abstract

Let G be a finite group and let $P\left(G\right)$ be the undirected power graph of G. Recall that the vertices of $P\left(G\right)$ are labelled by the elements of G, with an edge between $g_1$ and $g_2$ if either $g_1\in 〈g_2〉$ or $g_2\in 〈g_1〉$. The subgraph induced by the non-identity elements is called the reduced power graph, denoted by $P^{⁎}\left(G\right)$. The main purpose of this paper is to investigate the finite groups whose reduced power graph is claw-free, which means that it has no vertex with three pairwise non-adjacent neighbours. In particular, we prove that if $P^{⁎}\left(G\right)$ 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 $\mathrm{PSL}(2,q)$ for suitable choices of q. Finally we prove that if $P^{⁎}\left(G\right)$ is claw-free, then the order of G is divisible by at most 5 different primes.