The generating pairs of the 2-transitive groups

Abstract

Given a finite group $G$. The generating pair $(H,a)$ of $G$, that is, $H<G$ and $a\in G$ such that $⟨a,H⟩=G$. In this paper, we introduce the definition of FF-subgroup to characterize the generating pairs of the symmetric groups, alternating groups and projective groups $PSL(2,q)$. This gives a partial answer to an open problem of J. André and J. Araújo and P. J. Cameron.

7 pages