The prime graphs of some classes of finite groups

In this paper we study prime graphs of finite groups. The graph a group $G$, also known as the Gruenberg-Kegel graph, is with vertex set {primes dividing $|G|$} and an edge $p$-$q$ if only there exists element order $pq$ in $G$. theory, studying has been important topic for past almost half century. Only recently solvable groups have characterized theoretical terms only. paper, continue line research give complete characterizations several classes groups, including square-free order, metanilpotent cube-free and, any $n\in \mathbb{N}$, $n^\text{th}$-power-free order. We explore whose composition factors are cyclic or $A_5$ draw connections to conjecture Maslova. then propose algorithm that recovers from dual graph.