Permutation groups, partition lattices and block structures

TL;DR

This paper explores the structure of permutation groups with specific properties related to invariant partitions, block structures, and their algebraic and lattice-theoretic characteristics, extending classical theorems to new classes of groups.

Contribution

It introduces and studies the OB and PB properties of permutation groups, extending classical wreath product embedding theorems to these classes and analyzing their lattice and automorphism structures.

Findings

Groups with PB property can be embedded into generalized wreath products.

The lattice of invariant partitions can be distributive, modular, or orthogonal, affecting group structure.

The correspondence between posets and wreath products preserves intersections and inclusions.

Abstract

Let $G$ be a transitive permutation group on $\Omega$. The $G$-invariant partitions form a sublattice of the lattice of all partitions of $\Omega$, having the further property that all its elements are uniform (that is, have all parts of the same size). If, in addition, all the equivalence relations defining the partitions commute, then the relations form an \emph{orthogonal block structure}, a concept from statistics; in this case the lattice is modular. If it is distributive, then we have a \emph{poset block structure}, whose automorphism group is a \emph{generalised wreath product}. We examine permutation groups with these properties, which we call the \emph{OB property} and \emph{PB property} respectively, and in particular investigate when direct and wreath products of groups with these properties also have these properties. A famous theorem on permutation groups asserts that a transitive imprimitive group $G$ is embeddable in the wreath product of two factors obtained from the group (the group induced on a block by its setwise stabiliser, and the group induced on the set of blocks by~$G$). We extend this theorem to groups with the PB property, embeddng them into generalised wreath products. We show that the map from posets to generalised wreath products preserves intersections and inclusions. We have included background and historical material on these concepts.