%0 Journal Article
%T The classification of normalizing groups
%A Jo£¿o Ara¨²jo
%A Peter J. Cameron
%A James Mitchell
%A Max Neunh£¿ffer
%J Mathematics
%D 2012
%I arXiv
%X 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.
%U http://arxiv.org/abs/1205.0450v3