On abelian-by-cyclic Moufang loops (Q6123162)

A Moufang loop \(Q\) with an abelian normal subgroup \(X\) such that \(Q/X\) is a cyclic group is called an abelian-by-cyclic Moufang loop. In this paper, the authors study abelian-by-cyclic Moufang loops in a systematic way. The authors give a construction of abelian-by-cyclic Moufang loops from construction pairs, construct all split abelian-by-cyclic Moufang loops in which both \(X\) and \(Q/X\) are 3-divisible, and prove 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.