Orbit coherence in permutation groups

TL;DR

This paper explores the concept of orbit coherence in permutation groups, establishing conditions under which the set of orbit partitions forms a lattice and classifying primitive and normalizer groups with this property.

Contribution

It introduces orbit coherence, proves the centralizer's orbit partitions form a lattice under certain conditions, and classifies primitive and normalizer groups with join-coherence.

Findings

Centralizer of a permutation in symmetric group is meet-coherent.

Orbit partitions of centralizers form a lattice for finite sets.

Classification of primitive join-coherent groups and those normalizing an n-cycle.

Abstract

This paper introduces the notion of orbit coherence in a permutation group. Let $G$ be a group of permutations of a set $\Omega$. Let $\pi(G)$ be the set of partitions of $\Omega$ which arise as the orbit partition of an element of $G$. The set of partitions of $\Omega$ is naturally ordered by refinement, and admits join and meet operations. We say that $G$ is join-coherent if $\pi(G)$ is join-closed, and meet-coherent if $\pi(G)$ is meet-closed. Our central theorem states that the centralizer in $\Sym(\Omega)$ of any permutation $g$ is meet-coherent, and subject to a certain finiteness condition on the orbits of $g$, also join-coherent. In particular, if $\Omega$ is a finite set then the orbit partitions of elements of the centralizer in $\Sym(\Omega)$ of $g$ form a lattice. A related result states that the intransitive direct product and the imprimitive wreath product of two finite permutation groups are join-coherent if and only if each of the groups is join-coherent. We also classify the groups $G$ such that $\pi(G)$ is a chain and prove two further theorems classifying the primitive join-coherent groups of finite degree, and the join-coherent groups of degree $n$ normalizing a subgroup generated by an $n$-cycle.