When are inner mapping groups generated by conjugation maps?

摘要

A subloop of a loop Q is said to be normal if it is stabilized by all maps in the inner mapping group of Q. Here we show that in many cases, the inner mapping group of a Moufang loop is actually generated by conjugation maps. This includes any Moufang loop whose cubes generate either the entire loop or a subloop of index three. Such a result can be an extremely useful tool when proving that certain subloops are indeed normal just by showing that they are stabilized by the conjugation maps.