Moufang loops that share associator and three quarters of their multiplication tables

Abstract

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 $n\le 32$. 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.