On abelian-by-cyclic Moufang loops

We initiate a systematic study of abelian-by-cyclic Moufang loops, that is, Moufang loops Q with an abelian normal subgroup X such that Q / X is a cyclic group. Among other results, we construct all split abelian-by-cyclic Moufang loops in which both X and Q / X are 3-divisible, using so-called Moufang permutations on X, which are permutations that deviate from an automorphism of X by an alternating biadditive mapping. Additional abelian-by-cyclic Moufang loops are obtained from so-called construction pairs. As an aside, we show that, in a Moufang loop Q obtained from a construction pair, the abelian normal subgroup X induces an abelian congruence of Q if and only if Q is a group.