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 ≤ G and N ⊴ G then the group structure of G is determined by the subgroups H and N, the intersection N∩H and how H acts on N with a homomorphism φ : 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 N ∩ H, and how u permutes the elements of N as a semi-automorphism of N.