On the minimal degree of a primitive permutation group
A previous result of \textit{M. W. Liebeck} and \textit{J. Saxl} [Proc. Lond. Math. Soc., III. Ser. 63, No. 2, 266-314 (1991; Zbl 0696.20004)] concerning the minimal degree of a primitive permutation group is improved. The main result is the following. Let \(G\) be a primitive permutation group acting on a set \(\Omega\) of size \(n\). If \(x\in G\) fixes at least \(|\Omega|/2\) points in \(\Omega\) then one of the following holds. 1. \(G\) is of affine type and \(x\) acts as a transvection. 2. \(G\) acts as a subgroup of a wreath product in its product action with socle \({A_m}^r\) and \(n={m\choose k}^r\) or \(n=6^r\). 3. \(G\) acts as a subgroup of a wreath product in its product action with socle \(L^r\) where \(L\) is of orthogonal type \(O_m(2)\) with \(m\geq 6\). (In Theorem 1 more details are given.) This main result is used to strengthen a result of \textit{R. M. Guralnick} and \textit{M. G. Neubauer} [Contemp. Math. 186, 325-352 (1995; Zbl 0845.20002)] on generic covers of Riemann surfaces.
scientific article; zbMATH DE number 1213849