Quasigroups arisen by right nuclear extension

the aim of this paper is to prove that a quasigroup Q with right unit is isomorphic to an f-extension of a right nuclear normal subgroup G by the factor quasigroup Q/G if and only if there exists a normalized left transversal ∑ ⊂ Q to G in Q such that the right translations by elements of ∑ commute with all right translations by elements of the subgroup G. Moreover, a loop Q is isomorphic to an f-extension of a right nuclear normal subgroup G by a loop if and only if G is middle-nuclear, and there exists a normalized left transversal to G in Q contained in the commutant of G.