On finite groups whose power graph is claw-free

Let G be a finite group and let P(G) be the undirected power graph of G. Recall that the vertices of P(G) are labelled by the elements of G, with an edge between g1 and g2 if either g1∈〈g2〉 or g2∈〈g1〉. The subgraph induced by the non-identity elements is called the reduced power graph, denoted by P⁎(G). 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⁎(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.