The generating pairs of the 2-transitive groups

TL;DR

This paper introduces the concept of FF-subgroups to characterize generating pairs in certain finite groups, addressing an open problem in group theory related to symmetric, alternating, and projective groups.

Contribution

It defines FF-subgroups and uses them to characterize generating pairs in symmetric, alternating, and PSL(2,q) groups, providing partial answers to an open problem.

Findings

Characterization of generating pairs via FF-subgroups

Partial solution to an open problem in group theory

Application to symmetric, alternating, and PSL(2,q) groups

Abstract

Given a finite group $G$. The generating pair $(H,a)$ of $G$, that is, $H<G$ and $a\in G$ such that $\langle a,H\rangle=G$. 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. Andr\'e and J. Ara\'ujo and P. J. Cameron.