Transitive permutation groups with cyclic point stabilizers of maximum order

The purpose of this paper is to classify transitive permutation groups where the stabiliser is cyclic of order $n-1$ where the degree is $n-1$. This is a continuation of a paper by {\it A. Lucchini} [Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 9, No. 4, 241-243 (1998; Zbl 0940.20006)]. However the main result is more general, Theorem: Let $G$ be a finite group and let $\pi$ be a set of primes dividing the order of $G$. Assume that $G$ has an Abelian Hall $\pi$-subgroup, $Q$. Then one of the following hold: (a) $O_\pi(G)\neq 1$, (b) $|G|>2|Q|^2$, or (c) $G=E_1\times\cdots\times E_r$ where each $E_i$ is a $2$-transitive Frobenius group whose complement is a $\pi$-group.\par The question arose from a paper by {\it L. Babai}, {\it A. J. Goodman} and {\it L. Pyber} [J. Algebra 195, No. 1, 1-29 (1997; Zbl 0886.20020)]. [Alan R. Camina (Norwich)]