Moufang loops of class 2 and cubic forms

Abstract

We classify finite Moufang loops which are centrally nilpotent of class 2 in terms of certain cubic forms, concentrating on small Frattini Moufang loops, or SFMLs, which are Moufang loops L with a central subgroup Z of order p such that L/Z is an elementary abelian p-group. (For example, SFM 2-loops are precisely the class of code loops, in the sense of Griess.)More specifically, we first show that the nuclearly-derived subloop (normal associator subloop) of a Moufang loop of class 2 has exponent dividing 6. It follows that the subloop of elements of p-power order is associative for p > 3. Next, we show that if L is an SFML, then L/Z has the structure of a vector space with a symplectic cubic form. We then show that every symplectic cubic form is realized by some SFML and that two SFMLs are isomorphic in a manner preserving the central subgroup Z if and only if their symplectic cubic spaces are isomorphic up to scalar multiple. Consequently, we also obtain an explicit characterization of isotopy in SFM 3-loops. Finally, we extend many of our results to all finite Moufang loops of class 2.