The classification of normalizing groups

Araújo, João; Cameron, Peter J.; Mitchell, James D.; Max, Neunhöffer

http://hdl.handle.net/10400.2/3812

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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.