Araújo, JoãoBentz, WolframCameron, Peter2023-01-302023-01-302021-01-01J. Araújo, W. Bentz, and P.J. Cameron, “Primitive Permutation Groups and Strongly Factorizable Transformation SemiGroups”, Journal of Algebra 565 (2021), 513-530.0021-8693http://hdl.handle.net/10400.2/13255Preprint de J. Araújo, W. Bentz, and P.J. Cameron, “Primitive Permutation Groups and Strongly Factorizable Transformation Semigroups”, Journal of Algebra 565 (2021), 513-530.Let Ω be a finite set and T (Ω) be the full transformation monoid on Ω. The rank of a transformation t ∈ T (Ω) is the natural number |Ωt|. Given A ⊆ 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 ∈ T (Ω), every element in St := 〈G, t〉 can be written as a product eg, where e2 = e ∈ 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 s ∈ S there exists s′ ∈ S such that s = ss′s.) The paper ends with a list of problems.engjournal article10.1016/j.jalgebra.2020.05.0231090-266X