1996/11/20 by Hsu, Tim
#FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.math/9611214
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 Lp (the subloop of L of elements of p-power order) is associative for p>3. Next, a loop L is said to be a \it small Frattini Moufang loop, or SFML, if L has a central subgroup Z of order p such that C\isom L/Z is an elementary abelian p-group. C is thus given the structure of what we call a \it 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.