Polynomial clones on groups of order \(pq\)
\textit{P. M. Idziak} [Int. J. Algebra Comput. 9, No. 2, 213--226 (1999; Zbl 1023.08003)] had conjectured that, for any square-free \(n\), the cyclic group of order \(n\) has only finitely many polynomially inequivalent expansions. For \(n = pq\) where \(p\) and \(q\) are distinct primes, this conjecture is confirmed by the following results of the present paper: There are only finitely many constantive clones on a \(pq\)-element set that have some group operation among their binary operations, which follows from the fact that each such clone is uniquely determined by congruences and commutator relations. In particular, there are exactly 17 polynomially inequivalent expansions of the cyclic group of order \(pq\).
scientific article; zbMATH DE number 5150268
scientific article