Cyclic extensions of Moufang loops induced by semi-automorphisms

Abstract

It is well known that if a group G factorizes as G = NH where H\leq G and N is normal in G then the group structure of G is determined by the subgroups H and N, the intersection of N with H and how H acts on N with a homomorphism f : H -> Aut(N). Here we generalize the idea by creating extensions using the semi-automorphism group of N. We show that if G=NH is a Moufang loop, N is a normal subloop, and H=<u> is a finite cyclic group of order coprime to three then the binary operation of G depends only on the binary operation of N, the intersection of N with H, and how u permutes the elements of N as a semi-automorphism of N.