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}.     

Orbits of finite solvable groups on characters

We prove that if a solvable group A acts coprimely on a solvable group G, then A has a “large” orbit in its corresponding action on the set of ordinary complex irreducible characters of G. This extends (at the cost of a weaker bound) a 2005 result of A. Moretó who obtained such a bound in case that A is a p-group.     

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].     

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.     

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.