On the Coincidence of Gruenberg–Kegel Graphs of an Almost Simple Group and a Nonsolvable Frobenius Group

Let $$G$$ be a finite group. Its spectrum $$\omega(G)$$ is the set of all element orders . The prime $$\pi(G)$$ divisors order Gruenberg–Kegel graph (or graph) $$\Gamma(G)$$ simple with vertex in which any two vertices $$p$$ and $$q$$ are adjacent if only $$pq\in\omega(G)$$ structural theorem implies that class groups disconnected graphs widely generalizes Frobenius groups, whose role group theory absolutely exceptional. question coincidence an almost naturally arises. answer to known cases when solvable coincides its socle. In this short note we case nonsolvable socle isomorphic $$PSL_{2}(q)$$ for some