Equivalence and equation solvability problems for the alternating group

It is observed in this paper that the complexities of the equivalence and the equation solvability problems are not determined by the clone of the algebra. In particular, we prove that for the alternating group on four elements these problems have complexity in P; if we extend the group by the commutator as an extra operation, then the equivalence problem is coNP-complete and the equation solvability problem is NP-complete.