Primitive permutation groups and strongly factorizable transformation semigroups

Let Ω be a finite set and T(Ω) be the full transformation monoid on Ω. The rank of a transformation t in T(Ω) is the natural number |Ωt|. Given a subset A of T(Ω), denote by ⟨A⟩ the semigroup generated by A. Let k be a fixed natural number such that 2 ≤ k ≤ |Ω|. In the first part of this paper we (almost) classify the permutation groups G on Ω such that for all rank k transformations t in T(Ω), every element in St = ⟨G,t⟩ can be written as a product eg, where e is an idempotent in St and g∈G. In the second part we prove, among other results, that if S ≤ T(Ω) and G is the normalizer of S in the symmetric group on Ω, then the semigroup SG is regular if and only if S is regular. (Recall that a semigroup S is regular if for all x∈S there exists y∈S such that x = xyx.) The paper ends with a list of problems.