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Let $X$ be a finite set such that $|X|=n$. Let $\trans$ and $\sym$ denoterespectively 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 $G$ isnormalizing. The goal of this paper is to classify normalizing groups and henceanswer a question posed elsewhere. The paper ends with a number of problems forexperts in groups, semigroups and matrix theory.

The classification of normalizing groups