Permutation groups, partition lattices and block structures

TL;DR

This work analyzes when G-invariant partitions of a finite set Ω form structured lattices—orthogonal block structures (OBS) or poset block structures (PB)—and connects these lattices to group-theoretic constructions via generalised wreath products. It introduces OB and PB properties for transitive permutation groups, proves that PB groups embed into generalised wreath products, and generalizes the Krasner–Kaloujnine embedding to PB groups. A key contribution is showing that the map from posets to generalised wreath products preserves intersections and inclusions, unifying lattice-theoretic and group-theoretic perspectives while clarifying when modularity or distributivity arises. The paper also situates these concepts within the history of experimental design, links OB/PB structures to association schemes, and provides computational directions and open problems, highlighting practical implications for design of experiments and the study of permutation groups.

Abstract

Let $G$ be a transitive permutation group on $Ω$. The $G$-invariant partitions form a sublattice of the lattice of all partitions of $Ω$, 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.

Figures (6)

• Figure 1: The modular law for commuting partitions
• Figure 2: The lattices $P_5$ (left) and $N_3$ (right)
• Figure 3: Orthogonal block structures mentioned by Yates in FYCE
• Figure 4: More orthogonal block structures mentioned by Yates
• Figure 5: Some orthogonal block structures in ARL

Theorems & Definitions (33)

• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• Theorem 2.4
• Theorem 2.5
• Proposition 2.6
• Proposition 2.7
• Theorem 4.1
• Corollary 4.2
• Corollary 4.3