Abstract. We provide a necessary and sufficient condition on an element of a finite Coxeter group to ensure the transitivity of the Hurwitz action on its set of reduced decompositions into products of reflections. We show that this action is transitive if and only if the element is a parabolic quasi-Coxeter element, that is, if and only if it has a reduced decomposition into a product of reflections that generate a parabolic subgroup.

Abstract. In this note, we provide a short and self-contained proof that the braid group on n strands acts transitively on the set of reduced factorizations of a Coxeter element in a Coxeter group of finite rank n into products of reflections. We moreover use the same argument to also show that all factorizations of an element in a parabolic subgroup of W also lie in this parabolic subgroup.

This paper starts the classification of the primitive permutation groups (G, ) such that G contains a regular subgroup X. We determine all the triples (G, , X) with soc(G) an alternating, or a sporadic or an exceptional group of Lie type. Further, we construct all the examples (G, , X) with G a classical group which are known to us. Our particular interest is in the 8-dimensional orthogonal groups of Witt index 4. We determine all the triples (G, , X) with soc(G) ∼ = P + 8 (q). In order to obtain all these triples, we also study the almost simple groups G with G ∼ = P 2n+1 (q). The case G ∼ = U n (q) is started in this paper and finished in [B. Baumeister, Primitive permutation groups of unitary type with a regular subgroup, Bull. Belg. Math. Soc. 112 (5) (2006) 657-673]. A group X is called a Burnside-group (or short a B-group) if each primitive permutation group which contains a regular subgroup isomorphic to X is necessarily 2-transitive. In the end of the paper we discuss B-groups.

In this paper we lay the foundations for the study of permutation polytopes: the convex hull of a group of permutation matrices.We clarify the relevant notions of equivalence, prove a product theorem, and discuss centrally symmetric permutation polytopes. We provide a number of combinatorial properties of (faces of) permutation polytopes. As an application, we classify 4-dimensional permutation polytopes and the corresponding permutation groups. Classification results and further examples are made available online.We conclude with several questions suggested by a general finiteness result.

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