CLASS 2 MOUFANG LOOPS, SMALL FRATTINI MOUFANG LOOPS, AND CODE LOOPS

TIM HSU

Abstract. Let L be a Moufang loop which is centrally nilpotent of class 2. We first show that the nuclearly-derived subloop (normal associator subloop) L^{∗} of L has exponent dividing 6. It follows that L_{p} (the subloop of L of elements of p-power order) is associative for p > 3. Next, a loop L is said to be a small Frattini Moufang loop, or SFML, if L has a central subgroup Z of order p such that C ≅ L/Z is an elementary abelian p-group. C is thus given the structure of what we call a coded vector space, or CVS. (In the associative/group case, CVS’s are either orthogonal spaces, for p = 2, or symplectic spaces with attached linear forms, for p > 2.) Our principal result is that every CVS may be obtained from an SFML in this way, and two SFML’s are isomorphic in a manner preserving the central subgroup Z if and only if their CVS’s are isomorphic up to scalar multiple. Consequently, we obtain the fact that every SFM 2-loop is a code loop, in the sense of Griess, and we also obtain a relatively explicit characterization of isotopy in SFM 3-loops. (This characterization of isotopy is easily extended to Moufang loops of class 2 and exponent 3.) Finally, we sketch a method for constructing any finite Moufang loop which is centrally nilpotent of class 2.