Characterization of the group $G_2(5)$ by the prime graph - Darafsheh, M. R.; Nosratpour, P. (Ukrains’kyi Matematychnyi Zhurnal, 2016)
Let $G$ be a finite group. The prime graph of $G$ is a graph $Γ (G)$ with vertex set $π (G)$ and the set of all prime divisorsof $|G|$, where two distinct vertices $p$ and $q$ are adjacent by an edge if $G$ has an element of order $pq$. We prove that if$GΓ (G) = Γ (G_2(5))$, then $G$ has a normal subgroup $N$ such that $π (N) \subseteq \{ 2, 3, 5\}$ and $G/N \sim = G_2(5)$.