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.     

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.     

Coleman automorphisms of holomorphs of completely reducible and almost simple groups

Let H be the holomorph of a finite group G. It is proved that every Coleman automorphism of H is inner whenever G is either completely reducible or almost simple; in particular, this is the case when G is either characteristically simple or simple. As an application, we obtain the normalizer the conjecture holds for integral group rings of holomorphs of such groups in question.     

On orbital partitions and exceptionality of primitive permutation groups

Let and be transitive permutation groups on a set such that is a normal subgroup of . The overgroup induces a natural action on the set of non-trivial orbitals of on . In the study of Galois groups of exceptional covers of curves, one is led to characterizing the triples where fixes no elements of ; such triples are called exceptional. In the study of homogeneous factorizations of complete graphs, one is led to characterizing quadruples where is a partition of such that is transitive on ; such a quadruple is called a TOD (transitive orbital decomposition). It follows easily that the triple in a TOD is exceptional; conversely if an exceptional triple is such that is cyclic of prime-power order, then there exists a partition of such that is a TOD. This paper characterizes TODs such that is primitive and is cyclic of prime-power order. An application is given to the classification of self-complementary vertex-transitive graphs.

     

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.     

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.     

Extremely primitive groups and linear spaces

A non-regular primitive permutation group is called extremely primitive if a point stabilizer acts primitively on each of its nontrivial orbits. Let S be a nontrivial finite regular linear space and G ≤ Aut(S). Suppose that G is extremely primitive on points and let rank(G) be the rank of G on points. We prove that rank(G) ≥ 4 with few exceptions. Moreover, we show that Soc(G) is neither a sporadic group nor an alternating group, and G = PSL(2, q) with q + 1 a Fermat prime if Soc(G) is a finite classical simple group.