On finite groups whose power graph is claw-free

TL;DR

This paper characterizes finite groups with claw-free reduced power graphs, showing they are either solvable or almost simple with socle isomorphic to PSL(2,q), and limits the number of prime divisors of their order.

Contribution

It provides a complete classification of finite groups with claw-free reduced power graphs, linking group structure to graph-theoretic properties.

Findings

If the reduced power graph is claw-free, then the group is either solvable or almost simple.

For almost simple groups with claw-free reduced power graphs, the socle is isomorphic to PSL(2,q).

The order of such groups is divisible by at most five primes.

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 $g_1$ and $g_2$ if one of the two cyclic subgroups $\langle g_1\rangle, \langle g_2\rangle$ 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.