Orbits of Primitive $k$-Homogenous Groups on $(n-k)$-Partitions with Applications to Semigroups

TL;DR

This paper investigates the structure and properties of $k$-homogeneous groups, including their generation, orbits on partitions, and normalizers, and applies these findings to analyze transformation semigroups and their automorphisms.

Contribution

It provides new results on the generation, orbit classification, and normalizers of $k$-homogeneous groups, and applies these to transformation semigroups and automorphism analysis.

Findings

2-homogeneous groups are 2-generated

Classified $3$-homogeneous groups with invariant orbits

Described normalizers of 2-homogeneous groups

Abstract

Let $X$ be a finite set such that $|X|=n$, and let $k< n/2$. A group is $k$-homogeneous if it has only one orbit on the sets of size $k$. The aim of this paper is to prove some general results on permutation groups and then apply them to transformation semigroups. On groups we find the minimum number of permutations needed to generate $k$-homogeneous groups (for $k\ge 1$); in particular we show that $2$-homogeneous groups are $2$-generated. We also describe the orbits of $k$-homogenous groups on partitions with $n-k$ parts, classify the $3$-homogeneous groups $G$ whose orbits on $(n-3)$-partitions are invariant under the normalizer of $G$ in $S_n$, and describe the normalizers of $2$-homogeneous groups in the symmetric group. Then these results are applied to extract information about transformation semigroups with given group of units, namely to prove results on their automorphisms and on the minimum number of generators. The paper finishes with some problems on permutation groups, transformation semigroups and computational algebra.