Collapsing permutation groups

It is shown in [3] that any nonregular quasiprimitive permutation group is collapsing. In this paper we describe a wider class of collapsing permutation groups. Received June 6, 2000; accepted in final form August 11, 2000.     

TheL-sharp permutation groups

An answer is given to a problem proposed by Bannai and Ito for {I, I + s, I + s + t}-sharp permutation group, and the result is used to determineL-sharp groups for L={I, I + 1, I + 3} and {I, I + 2, I + 3}.     

Volterra series and permutation groups

Algebraic structures, connected with the asymptotic expansions of perturbations of smooth dynamical systems, are investigated; first of all, the so-called shuffle multiplication for permutations and for iterated integrals.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 39, pp. 3–40, 1991.     

Almost disjoint permutation groups

A permutation group on a set of (infinite) cardinality is almost disjoint if no element of except the identity has fixed points, i.e., if is an almost disjoint family of subsets of . We show how almost disjoint permutation groups can be constructed from almost disjoint families of sets.

     

The L-sharp permutation groups

An answer is given to a problem proposed by Bannai and Ito for {l,l+s,l+s+t}-sharp permutation group, and the result is used to determine L-sharp groups for L={l,l+1,l+3}and{l,l+2,l+3}.     

Partially ordered permutation groups

In this note we discuss permutation groups (G, ) in which the set admits aG-invariant order. By aG-invariant partial order (G-partial order) we mean a partial order < of such that < implies g<g, for all and in andg inG. If the set admits aG-partial order which is a total order, then (G, ) is an O-permutation group (orderable permutation group).The main concern of this paper is the development of a foundation for partially ordered permutation groups analogous to the existing one for partially ordered groups, as found in Fuchs [2].     

Barely transitive permutation groups

Part of this work was done while the author was the recipient of RGC Grant 1423 awarded by the University of Alabama Research Grants Committee.     

Orbit-equivalent infinite permutation groups

Let G,H be closed permutation groups on an infinite set X, with H a subgroup of G. It is shown that if G and H are orbit-equivalent, that is, have the same orbits on the collection of finite subsets of X, and G is primitive but not 2-transitive, then G=H.     

Descents, inversions, and major indices in permutation groups

A multivariate generating function involving the descent, major index, and inversion statistic first given by Ira Gessel is generalized to other permutation groups. We provide generating functions for variants of these three statistics for the Weyl groups of type B and D, wreath product groups, and multiples of permutations. All of our ideas are combinatorial in nature and exploit fundamental relationships between the elementary and homogeneous symmetric functions.     

Bases for permutation groups and matroids

In this paper, we give two equivalent conditions for the irredundant bases of a permutation group to be the bases of a matroid. (These are deduced from a more general result for families of sets.) If they hold, then the group acts geometrically on the matroid, in the sense that the fixed points of any element form a flat. Some partial results towards a classification of such permutation groups are given. Further, if G acts geometrically on a perfect matroid design, there is a formula for the number of G-orbits on bases in terms of the cardinalities of flats and the numbers of G-orbits on tuples. This reduces, in a particular case, to the inversion formula for Stirling numbers.