A criteria for a finite permutation group to be transitive

TL;DR

This paper proves a conjecture that any finite permutation group that is quasi-transitive on a finite set must also be transitive, establishing a clear criterion for transitivity.

Contribution

It confirms Camina's conjecture, showing that quasi-transitivity implies transitivity for finite permutation groups, a significant theoretical advancement.

Findings

Proves that quasi-transitivity implies transitivity in finite permutation groups.

Validates a previously conjectured criterion for transitivity.

Strengthens understanding of the structure of finite permutation groups.

Abstract

Let $G$ be a finite permutation group on a finite set $\Omega$. The notion of $G$ being quasi-transitive on $\Omega$ was defined by Alan Camina \cite{Camina}; in that paper conditions were established that ensured a quasi-transitive group on a finite set $\Omega$ was transitive on $\Omega$. The aim of this paper is to validate the conjecture made in \cite{Camina}: given any group $G$, if $G$ is quasi-transitive on a finite set $\Omega$ then $G$ is transitive on $\Omega$.