On finite groups whose power graph is claw-free

A graph is called claw-free if it contains no induced subgraph isomorphic to the complete bipartite graph K1, 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.