Let Ω be a finite set and T(Ω) the full transformation monoid on Ω. The rank of t∈T(Ω) is natural number |Ωt|. Given A⊆T(Ω), denote by 〈A〉 semigroup generated A. k fixed such that 2≤k≤|Ω|. In first part this paper we (almost) classify permutation groups G for all transformations t∈T(Ω), every element in St:=〈G,t〉 can written as product eg, where e2=e∈St g∈G. second prove, among other results, i...