A COMPARISON OF THE ORDER COMPONENTS IN FROBENIUS AND 2-FROBENIUS GROUPS WITH FINITE SIMPLE GROUPS

Abstract

Let $G$ be a finite group. Based on the Gruenberg-Kegel graph ${\rm GK}(G)$, the order of $G$ can be divided into a product of coprime positive integers. These integers are called the order components of $G$ and the set of order components is denoted by ${\rm OC}(G)$. In this article we prove that, if $S$ is a non-Abelian finite simple group with a disconnected graph ${\rm GK}(S)$, with an exception of $U_4(2)$ and $U_5(2)$, and $G$ is a finite group with ${\rm OC}(G)={\rm OC}(S)$, then $G$ is neither Frobenius nor $2$-Frobenius. For a group $S$ isomorphic to $U_4(2)$ or $U_5(2)$, we construct examples of $2$-Frobenius groups $G$ such that ${\rm OC}(S)={\rm OC}(G)$. In particular, the simple groups $U_4(2)$ and $U_5(2)$ are not recognizable by their order components.