The classification of normalizing groups

TL;DR

This paper classifies groups that normalize transformations outside the symmetric group, providing a comprehensive answer to a previously posed question and suggesting further research problems.

Contribution

It offers a complete classification of normalizing groups within the symmetric group, advancing understanding in group and semigroup theory.

Findings

Identifies all normalizing groups in the symmetric group

Provides criteria for a group to be normalizing

Suggests open problems for further research

Abstract

Let $X$ be a finite set such that $|X|=n$. Let $\trans$ and $\sym$ denote respectively the transformation monoid and the symmetric group on $n$ points. Given $a\in \trans\setminus \sym$, we say that a group $G\leq \sym$ is $a$-normalizing if $$<a,G> \setminus G=<g^{{-1}}ag\mid g\in G>.$$ If $G$ is $a$-normalizing for all $a\in \trans\setminus \sym$, then we say that $G$ is normalizing. The goal of this paper is to classify normalizing groups and hence answer a question posed elsewhere. The paper ends with a number of problems for experts in groups, semigroups and matrix theory.