THE EQUIVALENCE OF THE MOUFANG IDENTITIES: A SIMPLIFIED PROOF

G. G. Zaku and L.A. Ademola

Department of Mathematics, University of Jos, Jos, Nigeria.

Abstract

A Moufang loop M ,  is defined as a loop that satisfies any one of identities: xy  zx 

x  yzx , xy  zx  xyz  x, xy  zy  xy  zyor xy  xz  xy  xz .This definition assumes the

equivalence of these identities. The known proofs of the equivalence are cumbersome as

they require additional knowledge about autotopism and hence additional definitions

about mappings come into play. In this paper we provide an elegant alternative proof of

the equivalence-a proof that mainly uses clever manipulation of the Moufang identities

as well as the basic definitions of quasigroups, loops and the identity element.

Keywords: Moufang, loop, identity