On primitive permutation groups with nontrivial global stabilizers

In the paper the distinguishing number D(G) of an arbitrary finite primitive permutation group G is determined. As a consequence, the distinguishing number D(Г) of an arbitrary finite graph Г with a vertex-primitive group of automorphisms is found.     

On the maximal number of coprime subdegrees in finite primitive permutation groups

We compute the divisor class group of the general hypersurface Y of a complex projective normal variety X of dimension at least four containing a fixed base locus Z. We deduce that completions of normal local complete intersection domains of finite type over C of dimension ≥ 4 are completions of UFDs of finite type over C.     

Coprime subdegrees for primitive permutation groups and completely reducible linear groups

In this paper we answer a question of Gabriel Navarro about orbit sizes of a finite linear group H ? GL(V) acting completely reducibly on a vector space V: if the H-orbits containing the vectors a and b have coprime lengths m and n, we prove that the H-orbit containing a + b has length mn. Such groups H are always reducible if n,m > 1. In fact, if H is an irreducible linear group, we show that, for every pair of non-zero vectors, their orbit lengths have a non-trivial common factor. In the more general context of finite primitive permutation groups G, we show that coprime non-identity subdegrees are possible if and only if G is of O’Nan-Scott type AS, PA or TW. In a forthcoming paper we will show that, for a finite primitive permutation group, a set of pairwise coprime subdegrees has size at most 2. Finally, as an application of our results, we prove that a field has at most 2 finite extensions of pairwise coprime indices with the same normal closure.     

Note on primitive permutation groups and a diophantine equation

It is shown that there are no transitive rank 3 extensions of the projective linear groups H, PSL(m,q) ? H ? PFL(m,q), for any prime power q and integer m ? 3. In the course of the proof the diophantine equation 5^m + 11 = x^p2, where m, x are positive integers, arose. As such equations can now be solved completely we had the choice of using number theory or geometry to complete the proof.     

An upper bound for the order of primitive permutation groups

Let G be a primitive permutation group of order |G| and degree n. Then |G|≤nd^m, where d is the minimal size of a nontrivial orbit of a one-point stabilizer of G and m is the minimal degree of a nonprincipal irreducible representation of G entering its permutation representation. Bibliography: 8 titles. Dedicated to L. D. Faddeev on occasion of his 60th birthday Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 215, 1994, pp. 256–263. Translated by I. Ponomarenko.     

The primitive permutation groups of degree 2 a ·3 b

In this paper, we give a complete classification of all finite simple groups with maximal subgroups of index n, where n = 2^a·3^b for a, b≧ 1. As a consequence, for such n, all primitive permutation groups of degree n are given. The motivation of this work comes also from a study of Cayley graphs of certain valency on a finite simple group. Received: 9 March 2005