Class 2 Moufang loops, small Frattini Moufang loops, and code loops
Abstract
Let $L$ be a Moufang loop which is centrally nilpotent of class 2. We first show that the nuclearlyderived 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 {\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 2loop is a code loop, in the sense of Griess, and we also obtain a relatively explicit characterization of isotopy in SFM 3loops. (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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 November 1996
 DOI:
 10.48550/arXiv.math/9611214
 arXiv:
 arXiv:math/9611214
 Bibcode:
 1996math.....11214H
 Keywords:

 Mathematics  Group Theory