The classification of normalizing groups

Let X be a finite set such that |X|=n. Let Tn and Sn denote the transformation monoid and the symmetric group on n points, respectively. Given a∈Tn∖Sn, we say that a group G⩽Sn is a-normalizing if〈a,G〉∖G=〈g−1ag|g∈G〉, where 〈a,G〉 and 〈g−1ag|g∈G〉 denote the subsemigroups of Tn generated by the sets {a}∪G and {g−1ag|g∈G}, respectively. If G is a-normalizing for all a∈Tn∖Sn, then we say that G is normalizing.The goal of this paper is to classify the normalizing groups and hence answer a question of Levi, McAlister, and McFadden. The paper ends with a number of problems for experts in groups, semigroups and matrix theory.