A criteria for a finite permutation group to be transitive

Let $G$ be a finite permutation group on a finite set $Ω$. The notion of $G$ being quasi-transitive on $Ω$ was defined by Alan Camina \cite{Camina}; in that paper conditions were established that ensured a quasi-transitive group on a finite set $Ω$ was transitive on $Ω$. 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 $Ω$ then $G$ is transitive on $Ω$.

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