Finite-generation obstructions for classes of group isotopes

Let \\(G\\) be a nontrivial finite group and let \\(A\\) be a finite quasigroup of order \\(m\\). We construct an isotope of \\(G^{m!+1}\\) which violates an identity of \\(A\\). It follows that the isotope closure of the variety, quasivariety, or finite-algebra pseudovariety generated by \\(G\\) cannot be generated by a single finite quasigroup. For classes defined by disjunctive identities, we prove the corresponding negative assertion for the cyclic group of order three. These results answer Kourovka Problem 9.4 negatively under the respective conventions. The variety and quasivariety conclusions also follow from Falconer's earlier construction; the present argument uses finite separating isotopes.