Permutation groups and transformation semigroups: results and problems

TL;DR

This paper surveys recent results on transformation semigroups generated by permutation groups and a non-permutation element, exploring how properties like transitivity influence their structure and open problems in the field.

Contribution

It provides a comprehensive overview of how permutation group properties affect the structure of associated semigroups and discusses open problems linking group theory and semigroup theory.

Findings

Properties like regularity and idempotent generation are influenced by group characteristics.

The synchronization problem relates to the existence of rank 1 elements in the semigroup.

Open problems connect primitive permutation groups with semigroup structures.

Abstract

J.M. Howie, the influential St Andrews semigroupist, claimed that we value an area of pure mathematics to the extent that (a) it gives rise to arguments that are deep and elegant, and (b) it has interesting interconnections with other parts of pure mathematics. This paper surveys some recent results on the transformation semigroup generated by a permutation group $G$ and a single non-permutation $a$. Our particular concern is the influence that properties of $G$ (related to homogeneity, transitivity and primitivity) have on the structure of the semigroup. In the first part of the paper, we consider properties of $S=<G,a>$ such as regularity and idempotent generation. The second is a brief report on the synchronization project, which aims to decide in what circumstances $S$ contains an element of rank 1. The paper closes with a list of open problems on permutation groups and linear groups, and some comments about the impact on semigroups are provided. These two research directions outlined above lead to very interesting and challenging problems on primitive permutation groups whose solutions require combining results from several different areas of mathematics, certainly fulfilling both of Howie's elegance and value tests in a new and fascinating way.