An element a of a semigroup S is regular if it has a quasi-inverse, an element b such that aba=a and bab=b. A semigroup is regular if every element is regular.
João Araújo has recently proposed that advances in our knowledge of finite groups make it feasible to examine the general question: how does the structure of the group of units of a semigroup influence the structure of the semigroup? With James Mitchell and Csaba Schneider he showed that, if G is a subgroup of the symmetric group Sn, then ⟨G,a⟩ is a regular semigroup for all a in the full transformation semigroup Tn if and only if either G is the symmetric or alternating group, or G is one of nine specific groups of degree at most 9.
In more recent work, João and I have considered groups G for which a is a regular element of ⟨G,a⟩ for all elements a of Tn of rank k, for some fixed k. This condition is closely connected with two transitivity properties related to k-set transitivity. Using a combination of classical permutation group theory, the classification of finite simple groups, and computation, we have made some progress towards determining all such groups.