The automorphisms of a quasigroup or Latin square are permutations of the set of entries of the square, and thus belong to conjugacy classes in symmetric groups. These conjugacy classes may be recognized as being annihilated by symmetric group class functions that belong to a -ideal of the special -ring of symmetric group class functions.
The aim of this paper is to prove that a quasigroup with right unit is isomorphic to an -extension of a right nuclear normal subgroup by the factor quasigroup if and only if there exists a normalized left transversal to in such that the right translations by elements of commute with all right translations by elements of the subgroup . Moreover, a loop is isomorphic to an -extension of a right nuclear normal subgroup by a loop if and only if is middle-nuclear, and there exists...