Abstract:
   We study a construction introduced by Ales Drapal, giving rise to commutative A-loops of order kn where k and n are odd numbers. We show which combinations of k and n are possible if the construction is based on a field or on a quotient of the ring of integers. We conclude that if p and q are odd primes, there exists a non-associative commutative A-loop if and only if p divides q^2-1.