  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.
