Introduction. Let G be a compact connected Lie group and H a closed subgroup of G. Then the coset space G/H is a smooth manifold with G acting transitively as translations. It is easy to see that the natural action of G on G/H is effective if and only if H contains no nontrivial normal subgroup of G. For a given compact homogeneous space M=G/H, one might ask whether there are any other (differe...

An action of a group G is highly transitive if acts transitively on k-tuples distinct points for all k?1. Many examples groups with rich geometric or dynamical admit actions. We prove that admits such does not contain the subgroup finitary alternating permutations, and H confined G, then remains transitive, possibly after discarding finitely many points.

A permutation group is innately transitive if it has a transitive minimal normal subgroup, and this subgroup is called a plinth. In this paper we study three special types of inclusions of innately transitive permutation groups in wreath products in product action. This is achieved by studying the natural Cartesian decomposition of the underlying set that correspond to the product action of a w...

With the exception of Hering’s plane of order 27, all known odd order flag-transitive a‰ne planes are one of two types: admitting a cyclic transitive action on the line at infinity, or admitting a transitive action on the line at infinity with two equal-sized cyclic orbits. In this paper we show that when the dimension over the kernel for these planes is three, then the known examples are the o...

A generalization of some of the Folkman's constructions [13] of the so called semisymmetric graphs, that is regular graphs which are edgebut not vertex-transitive, was given in [22] together with a natural connection of graphs admitting 12 -arc-transitive group actions and certain graphs admitting semisymmetric group actions. This connection is studied in more detail in this paper. In Theorem 2...

We study the space H(d) of continuous Z-actions on the Cantor set, particularly questions on the existence and nature of actions whose isomorphism class is dense (Rohlin’s property). Kechris and Rosendal showed that for d = 1 there is an action on the Cantor set whose isomorphism class is residual; we prove in contrast that for d ≥ 2 every isomorphism class in H(d) is meager. On the other hand,...

A decomposition of a graph is a partition of the edge set, giving a set of subgraphs. A transitive decomposition is a decomposition which is highly symmetrical, in the sense that the subgraphs are preserved and transitively permuted by a group of automorphisms of the graph. This paper describes some ‘product’ constructions for transitive decompositions of graphs, and shows how these may be used...

A generalization of some of Folkman’s constructions (see (1967) J. Comb. Theory, 3, 215–232) of the so-called semisymmetric graphs, that is regular graphs which are edgebut not vertex-transitive, was given by Marušič and Potočnik (2001, Europ. J. Combinatorics, 22, 333–349) together with a natural connection between graphs admitting 1 2 -arc-transitive group actions and certain graphs admitting...