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 *S _{n}*, then ⟨

*G,a*⟩ is a regular semigroup for all

*a*in the full transformation semigroup

*T*if and only if either

_{n}*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 *T _{n}* 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.