The classification of finite simple Moufang loops

Abstract

The purpose of this paper is to classify the finite simple Moufang loops. A Moufang loop M is a loop which satisfies the identitynote that the equivalent identities ((xy)z)y = x(y(zy)), x(y(xz)) = ((xy)x)z also hold, by [2], p. 115. The Moufang loop M is simple if it has no non-trivial proper homomorphic images, or equivalently, if it has no non-trivial proper normal subloops. For basic definitions and properties of Moufang loops, see [2] – in particular, the Jordan–Hölder theorem holds for finite Moufang loops ([2], p. 67). Of course if the finite simple loop M is associative, then M is a simple group, and hence is determined by the classification of finite simple groups. In [9], Paige defines, for each finite field GF(q), a finite simple Moufang loop M(q) which is not associative – M(q) is essentially the set of units in the eight-dimensional split Cayley algebra over GF(q), modulo the centre (we shall describe M(q) in much more detail in §2).