Moufang Loops That Share Associator and Three Quarters of Their Multiplication Tables

Certain two constructions, due to Drápal, produce a group by modifying exactly one quarter of the Cayley table of another group. We present these constructions in a compact way, and generalize them to Moufang loops, using loop extensions. Both constructions preserve associators, the associator subloop, and the nucleus. We conjecture that two Moufang 2-loops of finite order n with equivalent associator can be connected by a series of constructions similar to ours, and offer empirical evidence that this is so for n = 16, 24, 32; the only interesting cases with n ≤ 32. We further investigate the way the constructions affect code loops and loops of type M(G, 2). The paper closes with several conjectures and research questions concerning the distance of Moufang loops, classification of small Moufang loops, and generalizations of the two constructions. MSC2000: Primary: 20N05. Secondary: 20D60, 05B15.