Characterization of the group $G_2(5)$ by the prime graph

Abstract

Let $G$ be a finite group. The prime graph of $G$ is a graph $\Gamma (G)$ with vertex set $\pi (G)$ and the set of all prime divisors of $|G|$, where two distinct vertices $p$ and $q$ are adjacent by an edge if $G$ has an element of order $pq$. We prove that if $G\Gamma (G) = \Gamma (G_2(5))$, then $G$ has a normal subgroup $N$ such that $\pi (N) \subseteq \{ 2, 3, 5\}$ and $G/N \sim = G_2(5)$.