\documentclass[12pt]{amsart} \headheight=6.5pt \textheight=574pt \textwidth=432pt \oddsidemargin=18pt \evensidemargin=18pt \topmargin=14pt \usepackage{amssymb} %%%%% Remove the next line if TeX complains about missing fonts, %%%%% %%%%% euscript.sty, etc. %%%%% \usepackage[mathcal]{euscript} %%%%% WARNING: HACK %%%%% %%%%% commented out %%%%% % Lots of big equations, so may be useful to allow display breaks % \allowdisplaybreaks[1] %%%%%%%%%% Preamble %%%%%%%%%% % Items done in default theorem style \newtheorem{thm}{Theorem}[section] \newtheorem{prop}[thm]{Proposition} \newtheorem{cor}[thm]{Corollary} \newtheorem{mainthm}{Main Theorem} \renewcommand{\themainthm}{\Alph{mainthm}} % Items done in definition style \theoremstyle{definition} \newtheorem{defn}[thm]{Definition} \newtheorem{rem}[thm]{Remark} \newtheorem{exmp}[thm]{Example} \newtheorem{open}[thm]{Question} \newtheorem*{notn}{Notation} \numberwithin{equation}{section} % Abbreviations \renewcommand{\a}{\alpha} \renewcommand{\b}{\beta} \renewcommand{\c}{\gamma} \renewcommand{\d}{\delta} \newcommand{\e}{\epsilon} \newcommand{\f}{\varphi} \renewcommand{\i}{^{-1}} \newcommand{\A}{\mathcal{A}} \newcommand{\B}{\mathcal{B}} \newcommand{\C}{\mathcal{C}} \newcommand{\D}{\mathcal{D}} \newcommand{\E}{\mathcal{E}} \newcommand{\F}{\mathbf{F}} \renewcommand{\P}{\mathcal{P}} \newcommand{\U}{\mathcal{U}} \newcommand{\V}{\mathcal{V}} \newcommand{\Z}{\mathbf{Z}} \renewcommand{\(}{\left(} \renewcommand{\)}{\right)} \renewcommand{\=}{\equiv} \newcommand{\adt}{\operatorname{adt}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\rad}{\operatorname{rad}} \renewcommand{\bar}[1]{\overline{#1}} \newcommand{\directsum}{\oplus} \newcommand{\dividing}{\mid} \newcommand{\gen}[1]{\left\langle #1 \right\rangle} \newcommand{\gilf}[1]{\left[ #1 \right]} \newcommand{\intersect}{\cap} \newcommand{\isom}{\cong} \newcommand{\lcm}{\operatorname{lcm}} \newcommand{\noqed}{\renewcommand{\qed}{}} \newcommand{\normal}{\vartriangleleft} \newcommand{\presentation}[2]{\gen{ {#1} \left| {#2} \right. }} \newcommand{\set}[1]{\left\{ {#1} \right\}} \newcommand{\tr}{\operatorname{trace}} \newcommand{\wt}[1]{\left| {#1} \right|} \newcommand{\sdcp}{% \,\mbox{{\footnotesize $\bigcirc$}\hspace*{-.715em}\raisebox{.05em}{\tiny $\pm$}}\mspace{4.9mu}} \newcommand{\cp}{\,\mbox{{\footnotesize $\bigcirc$}}\,} %%%%%%%%%% Document %%%%%%%%%% \begin{document} %%%%% intro, background \title[Class 2 and small Frattini Moufang loops]{Class 2 Moufang loops, small Frattini Moufang loops, and code loops} \author{Tim Hsu} \address{\hskip-\parindent Department of Mathematics \\ University of Michigan \\ Ann Arbor, MI 48109} \email{timhsu@@math.lsa.umich.edu} \subjclass{Primary 20N05; Secondary 20D15, 20D08} \keywords{Centrally nilpotent Moufang loops of class 2, Moufang $p$-loops, small Frattini Moufang loops, code loops} \date{\today} \begin{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 {\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. \end{abstract} \maketitle \section{Introduction} \label{sect:intro} The loops (groups without associativity) characterized by the near-associativity property \begin{equation} (xy)(zx) = x((yz)x) \end{equation} are known as {\it Moufang loops} (see Pflugfelder~\cite[Ch.\ IV]{hop:loops}). Many aspects of group theory may be generalized to Moufang loops, and among these aspects is the theory of {\it central nilpotence\/} (Bruck~\cite[Ch.\ VI]{rhb:loops}), the loop generalization of nilpotence in groups. Centrally nilpotent Moufang loops have been studied often, and provide many of the basic examples of finite Moufang loops. (See, for instance, Bruck~\cite[Ch.\ VIII, Thm.\ 10.1]{rhb:loops}, Chein~\cite[II.4]{oc:exloops}, Pflugfelder~\cite[Ch.\ IV]{hop:loops}, and Smith~\cite[p.\ 181]{jdhs:quasichar}.) Of particular relevance to this paper is the work of Glauberman and Wright~\cite{gg:oddloopsII,ggcrbw:nilploops}, who extended many of the standard theorems about finite nilpotent groups to finite centrally nilpotent Moufang loops. One class of centrally nilpotent Moufang loops which has particularly interesting applications to finite group theory is the class of {\it code loops}. The first code loop to be recognized as such was the {\it Parker loop\/} (named after its discoverer R. A. Parker), which played a key role in Conway's construction of the Monster finite simple group~\cite{jhc:monster}. Subsequently, Griess~\cite{rlg:codeloops} defined code loops to be certain central extensions of doubly even codes, providing the first published proof of their existence, and then went on~\cite{rlg:codetwolocal} to use code loops to construct 2-local subgroups of several other sporadic groups. For more on code loops and finite groups, including further references, see Griess~\cite{rlg:codetwolocal} and Richardson~\cite{tmr:oddcodeloops}. In this paper, we present some new results on centrally nilpotent Moufang loops of class 2, and apply these results by generalizing the theory of extraspecial groups (see, for instance, Aschbacher~\cite[Ch.\ 8]{ma:finite}) to {\it small Frattini Moufang loops}. For the convenience of the reader, we now summarize our main results. (The reader who is unfamiliar with the notation and terminology used here may first wish to read Section \ref{sect:background}.) Let $L$ be a Moufang loop which is centrally nilpotent of class 2, that is, a Moufang loop $L$ such that the quotient of $L$ by its center $Z(L)$ is an abelian group; and let $L_p$ be the set of all elements of $L$ whose order is a power of $p$. Recall that the nuclearly-derived subloop, or normal associator subloop, of $L$, which we denote by $L^*$, is the smallest normal subloop of $L$ such that $L/L^*$ is associative (is a group). Recall also that the torsion subloop (subloop of finite order elements) of $L$ is isomorphic to the (restricted) direct product of the subloops $L_p$, where $p$ runs over all primes (Thm.\ 6.2 of Bruck~\cite{rhb:somemoufang}, our Theorem \ref{thm:torsionsubloop}, or in the finite case, Cor.\ 1 of Glauberman and Wright~\cite{ggcrbw:nilploops}). In Section \ref{sect:classtwo}, we show that the commutator (resp.\ associator) function is a ``symplectic'' and ``multilinear'' function on $L/Z(L)\times L/Z(L)$ (resp.\ $L/Z(L)\times L/Z(L)\times L/Z(L)$) (Theorem \ref{thm:sympmulti}). As a consequence, we have: \begin{mainthm}\label{mainthm:assocsix} Let $L$ be a Moufang loop which is centrally nilpotent of class 2. Then $L^*$ (as defined above) has exponent dividing 6. In particular, for $p>3$, $L_p$ (as defined above) is associative (is a group). \end{mainthm} Compare the result of Bruck~\cite[VIII.2]{rhb:loops} that the cube of every associator of a commutative Moufang loop is trivial. (In fact, to prove Main Theorem \ref{mainthm:assocsix}, we use another case of the same formulas Bruck used to obtain that result.) We also note that Main Theorem \ref{mainthm:assocsix} is, in some sense, the best possible result of this type, since Example 3 of VII.5 of Bruck~\cite{rhb:loops} gives a construction of nonassociative finite Moufang $p$-loops of class 3 for all $p>2$. In Sections \ref{sect:eml}--\ref{sect:codedextensions}, we focus on {\it small Frattini Moufang loops\/} (also known as SFM loops, or SFML's), which are Moufang $p$-loops $L$ with a central subgroup $Z$ of order $p$ such that $C\isom L/Z$ is an elementary abelian group. (Note that we often think of $Z$ and $C$ as part of the structure of $L$.) SFML's are a class of Moufang loops often found ``in nature.'' For instance, every extraspecial Moufang loop (Definition \ref{defn:eml}) is an SFML. Also, every code loop is an SFM 2-loop, and conversely: \begin{mainthm}\label{mainthm:evenSFMLiscodeloop} Every SFM 2-loop is isomorphic to a code loop. \end{mainthm} Compare Thm.\ 14 of Griess~\cite{rlg:codeloops}, which shows that every loop constructed by ``Parker's procedure'' (Defn.\ 13 of Griess~\cite{rlg:codeloops}) is isomorphic to a code loop. For a more detailed comparison, see Remarks~\ref{rem:griesspf} and \ref{rem:codesize}. The key to Main Theorem \ref{mainthm:evenSFMLiscodeloop}, and also to Main Theorem \ref{mainthm:CVSisSFML}, below, is the fact that $L$ gives $C$ the structure of a {\it coded vector space}, or {\it CVS}, over $\F_p$. A CVS is a 4-tuple $(C,\sigma,\chi,\alpha)$, where $C$ is a vector space over $\F_p$, and $\sigma$, $\chi$, and $\alpha$ are ``symplectic 1-, 2-, and 3-forms'' on $C$ which are either multilinear, for $p>2$, or related by polarization, for $p=2$. The relationship between SFML's and CVS's can be stated as follows. \begin{mainthm}\label{mainthm:CVSisSFML} Every CVS can be obtained from some SFML in the manner described above. Furthermore, let $L$ and $M$ be SFML's, with distinguished central subgroups $Z_L$ and $Z_M$, and associated CVS's $C_L$ and $C_M$. Then there is an isomorphism $\phi: L \rightarrow M$ such that $\phi(Z_L)=Z_M$ if and only if $C_L$ and $C_M$ are isomorphic up to scalar multiple (action of $\Aut(Z_L)=\Aut(Z_M)$). \end{mainthm} The proofs of Main Theorems \ref{mainthm:evenSFMLiscodeloop} and \ref{mainthm:CVSisSFML} may be summarized as follows. \begin{list}{}{% \setlength{\leftmargin}{8em} \setlength{\labelwidth}{5em} \setlength{\labelsep}{\leftmargin} \addtolength{\labelsep}{-\labelwidth} \addtolength{\labelsep}{-\parindent} } \item[\bf Section \ref{sect:eml}.] We define the notion of a {\it coded extension\/} of a CVS (Definition \ref{defn:codedext}), and show that every SFML is a coded extension of a CVS, and vice versa (Theorem \ref{thm:codedextisSFML}). \item[\bf Section \ref{sect:cvsiscode}.] We show that every CVS over $\F_2$ can be obtained from a doubly even code (Theorems \ref{thm:codeisCVS} and \ref{thm:CVSiscode}). Main Theorem \ref{mainthm:evenSFMLiscodeloop} follows. \item[\bf Section \ref{sect:codedextensions}.] We show that every CVS has a unique coded extension (Theorems \ref{thm:codedextunique} and \ref{thm:codedextexist}). Main Theorem \ref{mainthm:CVSisSFML} follows. \end{list} We remark that the main technical tool used in Section \ref{sect:codedextensions}, the {\it semidirect central product}, is also useful for doing calculations in SFML's, especially code loops. In~\cite{th:codeloops}, we will address the general topic of decompositions of SFML's as semidirect central products of groups. In particular, we will give some short explicit constructions of code loops, including a Turyn-type construction for the Parker loop. For more details, see Remark~\ref{rem:sdcp}. Now, as mentioned above, every SFM 3-loop $L$ is a coded extension of some CVS over $\F_3$, say, $(C,\sigma(c),\chi(c,d),\alpha(c,d,e))$. For any $k\in C$, we define the {\it adjoint translate\/} $\adt_k(C)$ of $C$ to be the CVS $(C,\sigma(c),\chi(c,d)+\alpha(c,k,d),\alpha(c,d,e))$. A straightforward application of Main Theorem~\ref{mainthm:CVSisSFML} then gives the following characterization of isotopy in SFM 3-loops (Section \ref{sect:isotopy}). \begin{mainthm}\label{mainthm:isotopy} Let $L$ be a coded extension of a CVS $(C,\sigma,\chi,\alpha)$ over $\F_3$. Then up to isomorphism, the loop-isotopes of $L$ are precisely the coded extensions of the adjoint translates of $C$. In particular, $\sigma$ and $\alpha$ are ``isotopy invariants'' of $L$. \end{mainthm} Furthermore, since Main Theorem~\ref{mainthm:CVSisSFML} may be generalized directly to Moufang loops of class 2 and exponent 3, Main Theorem \ref{mainthm:isotopy} may also be extended to this situation. See Theorem \ref{thm:expthreeisotopy} for a precise statement. We conclude in Section \ref{sect:constructclasstwo} by giving a construction which can be used to obtain any finite Moufang loop of class 2. This construction generalizes much previous work, mostly in the commutative case. For instance, compare B\'{e}n\'{e}teau~\cite[IV.3]{lb:cml}, Bruck~\cite{rhb:contloops}, Chein~\cite[II.4]{oc:exloops}, and Ray-Chaudhuri and Roth~\cite{dkrcrr:hallcml}. \section{Background and notation} \label{sect:background} First, we set some conventions and notation to be used throughout. \begin{notn} Let $p$ be a prime. $\F_p$ denotes the field of order $p$, and $\F_p^\times$ its nonzero elements. Following group-theoretic custom, unless otherwise specified, we think of $\F_p$ as the group of order $p$, and $\F_p^\times$ as the automorphism group of $\F_p$. In this context, we identify the vector space $\F_p^k$ with the elementary abelian $p$-group of rank $k$, and we write vector addition in $\F_p^k$ multiplicatively, with the zero vector written as 1. If $a,b,c,\dots$ are elements or subsets of a loop (resp.\ vector space), $\gen{a,b,c,\dots}$ denotes the subloop (resp.\ subspace) generated by $a,b,c,\dots$. \end{notn} For those less familiar with loop theory, and for the purpose of establishing notation and terminology, we also review some definitions and results in loop theory, using Pflugfelder~\cite{hop:loops}, Bruck~\cite{rhb:loops}, and Chein, Pflugfelder, and Smith~\cite{cps:loopsappl} as our standard sources. \begin{defn} An {\it inverse property loop}, or in this paper, simply a {\it loop}, is a set $L$ with a binary operation (written as juxtaposition) having a unique identity element and unique two-sided inverses. (Note that the term ``inverse'' means that $a\i(ax)=(xa)a\i=x$.) In other words, a loop is a group minus associativity. \end{defn} Many concepts of group theory may be generalized to loop theory; we highlight the following ones. \begin{defn} For loop elements $\c$, $\d$, and $\e$, we define the {\it commutator\/} $[\c,\d]$ to be $(\d\c)\i(\c\d)$ and the {\it associator\/} $[\c,\d,\e]$ to be $(\c(\d\e))\i((\c\d)\e)$. In other words, \begin{align} \c\d &= (\d\c)[\c,\d], \\ (\c\d)\e &= (\c(\d\e))[\c,\d,\e], \\ \c(\d\e) &= ((\c\d)\e)[\c,\d,\e]\i. \end{align} (The inexperienced reader should note the inverse in the last formula.) \end{defn} \begin{defn}\label{defn:centnuc} Let $L$ be a loop. The {\it nucleus\/} of $L$ (denoted by $N(L)$) is the set of all $z\in L$ such that $[z,x,y]=[x,z,y]=[x,y,z]=1$ for all $x,y\in L$; and the {\it center\/} of $L$ (denoted by $Z(L)$) is defined to be the set of all $z\in N(L)$ such that $[z,x]=1$ for all $x,y\in L$. \end{defn} If $L$ is a loop, it can be shown (see Pflugfelder~\cite[I.3]{hop:loops}) that $N(L)$ is a subgroup of $L$, and that $Z(L)$ is an abelian subgroup of $N(L)$. \begin{defn} A {\it normal\/} subloop of a loop $L$ is any subloop of $L$ which is the kernel of some homomorphism from $L$ to a loop. \end{defn} For instance, any {\it central subgroup\/} (subgroup of $Z(L)$) of a loop $L$ is normal in $L$ (Pflugfleder~\cite[I.7]{hop:loops}). \begin{defn}\label{defn:derived} Let $L$ be a loop. We define the {\it centrally-derived subloop\/} (or normal commutator-associator subloop) of $L$ to be the smallest normal subloop $L'\normal L$ such that $L/L'$ is an abelian group. Similarly, we define the {\it nuclearly-derived subloop\/} (or normal associator subloop) of $L$ to be the smallest normal subloop $L^*\normal L$ such that $L/L^*$ is associative (is a group). \end{defn} See Bruck~\cite[Ch.\ VI]{rhb:loops} for a proof that $L'$ and $L^*$ are well-defined. Note that it follows from the isomorphism theorems for loops (see Pflugfelder~\cite[I.7]{hop:loops}) that $L'$ (resp.\ $L^*$) is the smallest normal subloop of $L$ containing all $[\c,\d]$ and $[\c,\d,\e]$ (resp.\ all $[\c,\d,\e]$), where $\c,\d,\e$ run over all elements of $L$. We will use Bruck's theory of central nilpotence~\cite[Ch.\ VI]{rhb:loops}, as described in the following definitions and Theorem \ref{thm:frattini}. \begin{defn} Let $L$ be a loop. The {\it upper central series\/} $\set{Z_i}$ of $L$ is defined by letting $Z_{0}=1$ and letting $Z_{i+i}$ be the unique subloop of $L$ containing $Z_{i}$ such that $Z_{i+i}/Z_{i}=Z(L/Z_{i})$. We say that $L$ is {\it centrally nilpotent of class $n$}, or simply of class $n$, if there exists $n$ such that $Z_{n}=L$ and $Z_{n-1}\neq L$. \end{defn} For instance, $L$ is of class 2 if and only if $L/Z(L)$ is an abelian group and $L$ is not, that is, if and only if $12$}. \end{cases} \label{eq:sigmalin} \end{align} \end{thm} \begin{proof} \eqref{eq:sigmapowerlin} is clear. As for \eqref{eq:sigmalin}, if $r=p(p-1)/2$, then \begin{equation} (\c\d)^p = \c^p \d^p \chi(d,c)^r = \sigma(c)\sigma(d)\chi(d,c)^r. \end{equation} The theorem follows from the fact that for $p>2$, $p$ divides $r$, and for $p=2$, $r=1$ and $\chi(c,d)=\chi(d,c)$. \end{proof} We are led to the following definition. \begin{defn}\label{defn:cvs} Let $Z$ be the group of order $p$. A {\it coded vector space\/} (or CVS) is defined to be a 4-tuple $(C,\sigma,\chi,\alpha)$, where $C$ is a finite-dimensional vector space over $\F_p$, and $\sigma: C\rightarrow Z$, $\chi: C\times C\rightarrow Z$, and $\alpha: C\times C\times C\rightarrow Z$ satisfy \eqref{eq:sigmapowerlin}--\eqref{eq:sigmalin}, \eqref{eq:chisymp}--\eqref{eq:chimultilin}, and \eqref{eq:alphasymp}--\eqref{eq:alphamultilin}, for all $c,d,e,f\in C$ and all $n\in\Z$. \end{defn} \begin{notn} We will often refer to the CVS $(C,\sigma,\chi,\alpha)$ simply as $C$. \end{notn} It is worth noting the different forms that \eqref{eq:sigmalin} and \eqref{eq:chimultilin} take for different $p$. That is, for $p=2$, we have \begin{align} \sigma(cd) &= \sigma(c)\sigma(d)\chi(c,d), \label{eq:sigmalineven} \\ \chi(cd,e) &= \chi(c,e)\chi(d,e)\alpha(c,d,e), \label{eq:chimultilineven} \end{align} and for $p>2$, we have \begin{align} \sigma(cd) &= \sigma(c)\sigma(d), \label{eq:sigmalinodd} \\ \chi(cd,e) &= \chi(c,e)\chi(d,e). \label{eq:chimultilinodd} \end{align} In other words, for $p=2$, $\sigma$, $\chi$, and $\alpha$ are related by polarization, and for $p>2$, $\sigma$ and $\chi$ are multilinear. As for $\alpha$, for $p=2$ or 3, $\alpha$ is multilinear, and for $p>3$, $\alpha$ is identically equal to 1. We also note that \eqref{eq:sigmapowerlin}, \eqref{eq:chipowerlin}, and \eqref{eq:alphapowerlin} imply \begin{equation}\label{eq:scaonone} \sigma(1)=\chi(c,1)=\alpha(c,d,1)=1 \end{equation} for all $c,d\in C$. Note that choosing a different generator for $Z$ has the effect of acting on $\sigma$, $\chi$, and $\alpha$ by an element of $\Aut(Z)$; in additive terms, this means that $C$ is really only defined up to scalar multiple. The natural definition of isomorphism for CVS's is therefore the following one. \begin{defn} Let $(C_i,\sigma_i,\chi_i,\alpha_i)$ be a CVS for $i=1,2$. We say that $C_1$ and $C_2$ are {\it isomorphic up to scalar multiple\/} if there is a vector space isomorphism $\phi: C_1 \rightarrow C_2$ and some fixed $a\in\Aut(Z)$ such that \begin{equation} \begin{split} \sigma_2(\phi(c)) &= \sigma_1(c)^a, \\ \chi_2(\phi(c),\phi(d)) &= \chi_1(c,d)^a, \\ \alpha_2(\phi(c),\phi(d),\phi(e)) &= \alpha_1(c,d,e)^a. \end{split} \end{equation} If $C_1$ and $C_2$ are isomorphic up to scalar multiple with respect to the trivial scalar (identity automorphism of $Z$), then we say that $C_1$ and $C_2$ are {\it isomorphic}. \end{defn} Finally, to describe the relationship between SFML's and CVS's, we introduce one more definition, in which, by convention, we define $\c^n$ inductively by $\c^0=1$ and $\c^{n+1}=\c\c^n$. \begin{defn}\label{defn:codedext} Let $p$ be a prime, and let $(C,\sigma,\chi,\alpha)$ be a CVS over $\F_p$. We say that a loop $L$ is a {\it coded extension\/} of $C$ if $L$ satisfies the following conditions. \begin{enumerate} \item\label{cond:centralext} $L$ has a central subgroup $Z$ of order $p$ such that $L/Z\isom C$. \item\label{cond:formulas} Let $\c,\d,\e\in L$ denote arbitrary preimages of $c,d,e\in C$, respectively. Then: \begin{align} \c^p &= \sigma(c), \label{eq:CEpower} \\ [\c,\d] &= \chi(c,d), \label{eq:CEcommute} \\ [\c,\d,\e] &= \alpha(c,d,e), \label{eq:CEassociate} \end{align} where the values of $\sigma$, $\chi$, and $\alpha$ are taken to be in the central subgroup $Z$. \end{enumerate} \end{defn} \begin{thm}\label{thm:codedextisSFML} Every SFML is a coded extension of a CVS, and every coded extension of a CVS is an SFML. \end{thm} \begin{proof} If $L$ is a Moufang loop with a central subgroup $Z$ of order $p$ such that $C\isom L/Z$ is an elementary abelian $p$-group, Theorems \ref{thm:sympmulti} and \ref{thm:linear} imply that $L$ is a coded extension of the CVS $(C,\sigma,\chi,\alpha)$, where $\sigma$, $\chi$, and $\alpha$ are from Definitions \ref{defn:chialpha} and \ref{defn:sigma}. Conversely, let $L$ be a coded extension of a CVS $(C,\sigma,\chi,\alpha)$. Because $\alpha$ satisfies \eqref{eq:alphasymp}--\eqref{eq:alphamultilin}, the proof of Theorem~\ref{thm:LisMklaw} in the case $k=1$ shows that $L$ is Moufang. (Note that for $k=1$, the proof of Theorem~\ref{thm:LisMklaw} does not use power-associativity.) Therefore, since $L/Z$ is an elementary abelian $p$-group, $L$ is an SFML. \end{proof} \section{Coded vector spaces and doubly even codes} \label{sect:cvsiscode} We come to the question: Given an $m$-dimensional vector space $C$ over $\F_p$, in what ways can $\sigma$, $\chi$, and $\alpha$ be defined to obtain a CVS? Now, for $p>2$, \eqref{eq:sigmalinodd} and \eqref{eq:chimultilinodd} show that $\sigma$, $\chi$, and $\alpha$ can be chosen independently, which makes this question easy. On the other hand, for $p=2$, $\sigma$, $\chi$, and $\alpha$ are related by polarization, so it is less clear {\it a priori\/} which CVS's exist over $\F_2$. Let $C$ be a CVS over $\F_2$, and let $\set{c_1\dots c_m}$ be a basis for $C$. Clearly, the symplectic, skew-symmetric (or in characteristic 2, symmetric), and polarization properties of $\sigma$, $\chi$, and $\alpha$ imply that $\sigma$, $\chi$, and $\alpha$ are determined by $\sigma(c_i)$ ($1\le i\le m$), $\chi(c_i,c_j)$ ($1\le i0$, choose elements $\sigma_i$ ($1\le i\le m$), $\chi_{ij}$ ($1\le i3$, we are in the associative case, which means that this is essentially known. However, to gain insight into the known cases and to introduce the {\it semidirect central product\/} (Definition~\ref{defn:sdcp}), we continue to consider all primes $p$. \begin{notn} In this section, we use script letters $\C,\D,\E,\dots$ to denote coded extensions of the CVS's $C,D,E,\dots$, possibly with subscripts. \end{notn} We first address uniqueness. Our proof follows \S2 of Conway~\cite{jhc:monster}. \begin{thm}\label{thm:codedextunique} For $n=1,2$, let $C_n$ be a CVS of dimension $k$ over $\F_p$, and let $\C_n$ be a coded extension of $C_n$. If $C_1$ and $C_2$ are isomorphic up to scalar multiple, then $\C_1$ is isomorphic to $\C_2$. \end{thm} \begin{proof} First, by choosing a different generator for the distinguished central subgroup of $\C_2$, we may assume that $C_1$ and $C_2$ are isomorphic. So let $\set{c_i}$ be a basis for $(C,\sigma,\chi,\alpha)=C_1$, and let $\C$ be the loop given by the loop presentation \begin{align} \left\langle\vphantom{\beta^{-1}}\right. z, \gamma_i \left.\vphantom{\beta^{-1}}\right| \qquad\qquad& \notag \\ \c_i^p &= \sigma(c_i), \label{eq:Cpowerreln} \\ [\c,\d] &= \chi(c,d), \label{eq:Ccommreln} \\ [\c,\d,\e] &= \alpha(c,d,e), \label{eq:Cassocreln} \\ z^p &= 1 \label{eq:zpower} \left.\vphantom{\beta^{-1}}\right\rangle, \end{align} where $i$ runs between $1$ and $k$; $\c,\d,\e$ run over all loop words in the generators; $c,d,e$ are the images in $C$ of $\c,\d,\e$, respectively, under the map sending $\c_i$ to $c_i$ and $z$ to 1; the values of $\sigma$, $\chi$, and $\alpha$ are taken to be in $Z=\gen{z}$, which is a central subgroup of $\C$ because of \eqref{eq:Ccommreln}--\eqref{eq:zpower} and \eqref{eq:scaonone}; and the expression $\c_i^p$ is defined inductively by $\c_i^0=1$ and $\c_i^{n+1}=\c_i\c_i^n$. Now, since isomorphisms take bases to bases, the above presentation is purely a function of the isomorphism class of $C$. Therefore, it is enough to show that any coded extension of $C$ is isomorphic to $\C$. Furthermore, since the universal property of loop presentations (see Evans~\cite[I.2]{te:varloops}) implies that {\em any\/} coded extension of $C$ is a homomorphic image of $\C$, it is enough to show that $\C$ has order at most $p^{1+k}$. However, since \eqref{eq:Ccommreln}--\eqref{eq:zpower} imply that $Z$ is a central subgroup of $\C$ of order dividing $p$, and \eqref{eq:Cpowerreln}--\eqref{eq:Cassocreln} implies that $\C/Z$ is an elementary abelian $p$-group of rank $k$, the theorem follows. \end{proof} \begin{notn} We resume the convention that if $\c,\d,\e$ (possibly with subscripts) are elements of a coded extension, then $c,d,e$ (possibly with subscripts) are the corresponding elements of the quotient CVS. \end{notn} We turn to existence. Now, if $\D$ and $\E$ are subloops of the same coded extension, then for $\d_i\in\D$, $\e_i\in\E$, $(\d_1\e_1)(\d_2\e_2)=z_0(\d_1\d_2)(\e_1\e_2)$, where $z_0\in Z$ is expressible in terms of $\chi$, $\alpha$, $d_1$, $e_1$, $d_2$ and $e_2$. This observation motivates the following defintion. \begin{defn}\label{defn:sdcp} Let $p$ be a prime and let $Z=\F_p$ be the group of order $p$. Let $D$ and $E$ be linearly independent subspaces of a CVS $(C,\sigma,\chi,\alpha)$; note that $D$ and $E$ are CVS's by restriction. Let $\D$ (resp.\ $\E$) be a coded extension of $D$ (resp.\ $E$). It is easily verified that the binary operation on the set $Z\times\D\times\E$ given by \begin{equation}\label{eq:semicentral} (z_1,\d_1,\e_1)(z_2,\d_2,\e_2) = (z_0 z_1 z_2,\d_1\d_2,\e_1\e_2), \end{equation} where \begin{equation}\label{eq:sdcperror} z_0 = \chi(e_1,d_2) \alpha(d_1,e_1 d_2\i,e_2) \alpha(d_1,e_1,d_2)^2 \alpha(e_1,d_2,e_2)^{-2}, \end{equation} defines a loop $\Gamma$ containing the central subgroup $Z\times Z\times Z$. We may therefore define the {\it semidirect central product\/} of $\D$ and $\E$ to be the quotient of $\Gamma$ by the central subgroup $\gen{(z,z\i,1),(z,1,z\i)}$, where $z$ runs over all elements of $Z$. \end{defn} Note that for $p=2$, \eqref{eq:sdcperror} becomes \begin{equation}\label{eq:semicentraleven} z_0 = \chi(e_1,d_2) \alpha(d_1,e_1 d_2,e_2), \end{equation} for $p=3$, \eqref{eq:sdcperror} becomes \begin{equation}\label{eq:semicentralodd} z_0 = \chi(e_1,d_2) \alpha(d_1,e_1 d_2\i,e_2) \alpha(d_1,e_1,d_2)\i \alpha(e_1,d_2,e_2), \end{equation} and for $p>3$, \begin{equation}\label{eq:semicentralbigp} z_0 = \chi(e_1,d_2). \end{equation} \begin{notn} Imitating the ATLAS~\cite{ATLAS} notation $\D\cp\E$ for the central product of two groups $\D$ and $\E$, we use $\D\sdcp\E$ to denote the semidirect central product (``central product up to sign'') of two code loops $\D$ and $\E$. Note that if the $\chi$ and $\alpha$ factors on the right-hand side of \eqref{eq:semicentral} are always $1$, then $\D\sdcp\E$ just becomes the ordinary central product. In this situation, we write $\D\cp\E$, just as in the group case. \end{notn} \begin{thm}\label{thm:sdcpiscodedext} Let $D$ and $E$ be linearly independent sub-CVS's of $(C,\sigma,\chi,\alpha)$, let $\D$ (resp.\ $\E$) be a coded extension of $D$ (resp.\ $E$), and let $\C=\D\sdcp\E$. Then $\C$ is a coded extension of $D\oplus E$. \end{thm} Throughout the following proof, by convention, $\c=(z,\d,\e)$, possibly with subscripts. (Note that by our usual convention, we then have $c=de$.) We also freely identify $Z$ with $Z\times 1\times 1$. Finally, the ``skew-symmetric,'' ``symplectic,'' and ``multilinear'' properties of $\chi$ and $\alpha$ (\eqref{eq:chisymp}--\eqref{eq:alphamultilin}) will be applied freely. \begin{proof} First, taking the quotient of $Z\times Z\times Z$ in $\C$ as our distinguished central subgroup, it is easy to see that condition \ref{cond:centralext} of Definition~\ref{defn:codedext} holds in $\C$. Furthermore, by collecting ``signs'' (elements of $Z$), it is easy to see that \eqref{eq:CEpower}--\eqref{eq:CEassociate} hold ``up to sign,'' so it remains to check the signs. We first verify \eqref{eq:CEpower}, recalling our convention that $\c^n$ is defined inductively by $\c^0=1$ and $\c^{n+1}=\c\c^n$. First, we claim that for all $n\ge 0$, \begin{equation}\label{eq:powerind} \c^n = (z^n\chi(e,d)^r,\d^n,\e^n) \end{equation} where $r=n(n-1)/2$. In fact, if \eqref{eq:powerind} holds, then \begin{equation} \begin{split} \c^{n+1} &= \c\c^n \\ &= (z,\d,\e)(z^n\chi(e,d)^r,\d^n,\e^n) \\ &= (z^{n+1}\chi(e,d)^r\chi(e,d^n),\d^{n+1},\e^{n+1}) \\ &= (z^{n+1}\chi(e,d)^{r+n},\d^{n+1},\e^{n+1}), \end{split} \end{equation} and since $r+n=n(n+1)/2$, \eqref{eq:powerind} follows by induction. In particular, if $n=p$, $r=p(p-1)/2$, which means that $\chi(e,d)^r=\chi(d,e)$ for $p=2$ and $\chi(e,d)^r=1$ for $p>2$. Therefore, collecting signs, we get \begin{equation} \begin{split} \c^p &= (\chi(e,d)^r,\sigma(d),\sigma(e)) \\ &= \begin{cases} \sigma(d)\sigma(e)\chi(d,e) &\text{for $p=2$} \\ \sigma(d)\sigma(e) &\text{for $p>2$} \end{cases} \\ &= \sigma(de) \\ &= \sigma(c). \end{split} \end{equation} Next, turning to \eqref{eq:CEcommute}, by collecting signs, we see that \begin{equation} \c_1\c_2 = (z_3 z_1 z_2,\d_1\d_2,\e_1\e_2), \end{equation} where \begin{equation} z_3 = \chi(e_1,d_2) \alpha(d_1,e_1 d_2\i,e_2) \alpha(d_1,e_1,d_2)^2 \alpha(e_1,d_2,e_2)^{-2}; \end{equation} and \begin{equation} \c_2\c_1 = (z_4 z_1 z_2,\d_1\d_2,\e_1\e_2), \end{equation} where \begin{equation}\label{eq:zfour} \begin{split} z_4 &= \chi(e_2,d_1) \chi(d_2,d_1) \chi(e_2,e_1) \\ &\qquad \alpha(d_2,e_2 d_1\i,e_1) \alpha(d_2,e_2,d_1)^2 \alpha(e_2,d_1,e_1)^{-2}. \end{split} \end{equation} To obtain \eqref{eq:CEcommute}, we need to show that $z = z_3 z_4\i = \chi(d_1 e_1,d_2 e_2)$. However, gathering the $\chi$ terms of $z$, we have \begin{equation} \begin{split} &\chi(e_1,d_2)\chi(e_2,d_1)\i\chi(d_2,d_1)\i\chi(e_2,e_1)\i \\ &\qquad = \chi(e_1,d_2)\chi(d_1,e_2)\chi(d_1,d_2)\chi(e_1,e_2) \\ &\qquad = \chi(d_1,d_2 e_2)\chi(e_1,d_2 e_2) \alpha(d_1,d_2,e_2)^3 \alpha(e_1,d_2,e_2)^3 \\ &\qquad = \chi(d_1 e_1,d_2 e_2) \alpha(d_1,e_1,d_2 e_2)^3 \alpha(d_1 e_1,d_2,e_2)^3, \end{split} \end{equation} and gathering the $\alpha$ terms of $z$, we have \begin{equation} \begin{split} & \alpha(d_1,e_1 d_2\i,e_2) \alpha(d_1,e_1,d_2)^2\alpha(e_1,d_2,e_2)^{-2} \\ {}\cdot{} & \alpha(d_2,e_2 d_1\i,e_1)\i \alpha(d_2,e_2,d_1)^{-2}\alpha(e_2,d_1,e_1)^2 \\ {}={} & \alpha(d_1,e_1,d_2)^3\alpha(d_1,e_1,e_2)^3 \alpha(d_1,d_2,e_2)^3\alpha(e_1,d_2,e_2)^3. \end{split} \end{equation} \eqref{eq:CEcommute} follows because $\alpha^6=1$ identically. Finally, using the same strategy to verify \eqref{eq:CEassociate}, we see that $(\c_1\c_2)\c_3 = z \cdot \c_1(\c_2\c_3)$, where \begin{equation*} \begin{split} z {}={} & \chi(e_1,d_2) \alpha(d_1,e_1 d_2\i,e_2) \alpha(d_1,e_1,d_2)^2 \alpha(e_1,d_2,e_2)^{-2} \\ {}\cdot{} & \chi(e_1 e_2,d_3) \alpha(d_1 d_2,e_1 e_2 d_3\i,e_3) \alpha(d_1 d_2,e_1 e_2,d_3)^2 \alpha(e_1 e_2,d_3,e_3)^{-2} \\ {}\cdot{} & \chi(e_2,d_3)\i \alpha(d_2,e_2 d_3\i,e_3)\i \alpha(d_2,e_2,d_3)^{-2} \alpha(e_2,d_3,e_3)^2 \\ {}\cdot{} & \chi(e_1,d_2 d_3)\i \alpha(d_1,e_1 d_2\i d_3\i,e_2 e_3)\i \alpha(d_1,e_1,d_2 d_3)^{-2} \alpha(e_1,d_2 d_3,e_2 e_3)^2 \\ {}\cdot{} & \alpha(d_1,d_2,d_3)\alpha(e_1,e_2,e_3). \end{split} \end{equation*} Collecting the $\chi$ terms of $z$, we have \begin{equation}\label{eq:chitermsofalpha} \begin{split} &\chi(e_1,d_2)\chi(e_1 e_2,d_3)\chi(e_2,d_3)\i\chi(e_1,d_2 d_3)\i \\ &\qquad \begin{split} {}={} &\chi(e_1,d_2)\chi(e_1,d_3)\chi(e_2,d_3) \alpha(e_1,e_2,d_3)^3 \\ &\chi(e_2,d_3)\i\chi(e_1,d_2)\i\chi(e_1,d_3)\i \alpha(e_1,d_2,d_3)^3 \end{split} \\ &\qquad = \alpha(e_1,e_2,d_3)^3 \alpha(e_1,d_2,d_3)^3. \end{split} \end{equation} Completely expanding all $\alpha$ terms of $z$ and collecting all terms which repeat subscripts, we have \begin{equation}\label{eq:magiccancel} \begin{split} %% (12)3 terms & \alpha(d_1,e_1,e_2)\alpha(d_1,d_2\i,e_2) \alpha(d_1,e_1,d_2)^2\alpha(e_1,d_2,e_2)^{-2} \\ {}\cdot{} & \alpha(d_1,d_3\i,e_3)\alpha(d_2,d_3\i,e_3) \alpha(d_1,e_1,e_3)\alpha(d_2,e_2,e_3) \\ {}\cdot{} & \alpha(d_1,e_1,d_3)^2\alpha(d_2,e_2,d_3)^2 \alpha(e_1,d_3,e_3)^{-2}\alpha(e_2,d_3,e_3)^{-2} \\[1ex] %% space has been inserted to separate two cancelling halves of %% equation %% 1(23) terms {}\cdot{} & \alpha(d_2,e_2,e_3)\i\alpha(d_2,d_3\i,e_3)\i \alpha(d_2,e_2,d_3)^{-2}\alpha(e_2,d_3,e_3)^2 \\ {}\cdot{} & \alpha(d_1,e_1,e_2)\i\alpha(d_1,e_1,e_3)\i \alpha(d_1,d_2\i,e_2)\i\alpha(d_1,d_3\i,e_3)\i \\ {}\cdot{} & \alpha(d_1,e_1,d_2)^{-2}\alpha(d_1,e_1,d_3)^{-2} \alpha(e_1,d_2,e_2)^2\alpha(e_1,d_3,e_3)^2 = 1, \end{split} \end{equation} since (thankfully) every term in the first three lines of \eqref{eq:magiccancel} is cancelled by some term in the next three lines. The remaining $\alpha$ terms of $z$ are then, after expansion, \begin{equation}\label{eq:allsubs} \begin{split} & \alpha(d_1,e_2,e_3)\alpha(d_2,e_1,e_3) \alpha(d_1,e_2,d_3)^2\alpha(d_2,e_1,d_3)^2 \\ {}\cdot{} & \alpha(d_1,d_2\i,e_3)\i\alpha(d_1,d_3\i,e_2)\i \alpha(e_1,d_2,e_3)^2\alpha(e_1,d_3,e_2)^2 \\ {}\cdot{} & \alpha(d_1,d_2,d_3)\alpha(e_1,e_2,e_3) \\ {}={} & \alpha(d_1,e_2,e_3)\alpha(e_1,d_2,e_3)\i \alpha(d_1,e_2,d_3)^2\alpha(e_1,d_2,d_3)^{-2} \\ {}\cdot{} & \alpha(d_1,d_2,e_3)\alpha(d_1,e_2,d_3)\i \alpha(e_1,d_2,e_3)^2\alpha(e_1,e_2,d_3)^{-2} \\ {}\cdot{} & \alpha(d_1,d_2,d_3)\alpha(e_1,e_2,e_3) \\ {}={} & \alpha(d_1,e_2,e_3)\alpha(d_1,e_2,d_3) \alpha(e_1,d_2,d_3)^{-2} \\ {}\cdot{} & \alpha(d_1,d_2,e_3)\alpha(e_1,d_2,e_3) \alpha(e_1,e_2,d_3)^{-2} \\ {}\cdot{} & \alpha(d_1,d_2,d_3)\alpha(e_1,e_2,e_3). \end{split} \end{equation} Multiplying \eqref{eq:chitermsofalpha}--\eqref{eq:allsubs}, we see that $z=\alpha(d_1 e_1,d_2 e_2,d_3 e_3)$, so \eqref{eq:CEassociate}, and the theorem, follow. \end{proof} \begin{rem} Compare Kitazume~\cite{mk:codeloops}, whose Thm.\ 2 is a particular example of the $p=2$ case of the above theorem. \end{rem} With Theorem \ref{thm:sdcpiscodedext}, to obtain the existence of coded extensions, we just need the following example. \begin{exmp}\label{exmp:dimone} Let $(C,\sigma,1,1)$ be a CVS of dimension 1, and let $c$ be a nonzero vector in $C$. If $\sigma(c)=1$, then $Z\times C$ is a coded extension of $C$; otherwise, the cyclic group of order $p^2$ is a coded extension of $C$. \end{exmp} \begin{thm}\label{thm:codedextexist} If $C$ is a CVS, there is a (unique) coded extension of $C$. \end{thm} \begin{proof} Proceeding by induction on $k=\dim C$, let $C=D+E$, where $\dim D=k-1$ and $\dim E=1$, let $\D$ be the coded extension of $D$ (by induction), and let $\E$ be the coded extension of $E$ (from Example \ref{exmp:dimone}). Then from Theorem \ref{thm:sdcpiscodedext}, $\D\sdcp\E$ is a coded extension of $C$. \end{proof} \begin{rem}\label{rem:sdcp} Consider again the loop $\C$ given by the presentation in the proof of Theorem \ref{thm:codedextunique}. Now, there is an easy ``solution'' to the loop word problem for $\C$. Namely, given a loop word $w$ in the generators $\set{\c_i}$, $w$ can be arranged into the normal form $z\c_1^r(\c_2^s(\c_3^t(\dots)))$ by simply powering, commuting, and associating elements, while keeping track of ``error terms'' with $\sigma$, $\chi$, and $\alpha$. Theorem \ref{thm:codedextexist} can then be interpreted as saying precisely that this solution is actually consistent, i.e., that we are never forced to kill $Z$. In fact, this normal form procedure provides a method for doing calculations in any SFML. Furthermore, while building an SFML from 1-dimensional pieces becomes unwieldy for large dimension, by using larger associative (or at least familiar) pieces as building blocks, and gluing them together with the semidirect central product, it is not hard to do hand calculations in, say, 12- or 16-dimensional examples. See~\cite{th:codeloops}. \end{rem} \begin{rem}\label{rem:conway} Conway (personal communication) has remarked that he and Par\-ker orignally verified the existence of the Parker loop (unpublished) by proving a ``consistency'' theorem analogous to MacLane's theorems for monoidal categories~\cite[Ch.\ VII]{sm:categories}, using ``pentagonal'' and ``hexagonal'' relations in the Parker loop. This approach could also probably be used to prove Theorem \ref{thm:codedextexist}. \end{rem} Finally, we prove Main Theorem \ref{mainthm:CVSisSFML}. \begin{proof}[{\bf Proof of Main Theorem \ref{mainthm:CVSisSFML}}] Because of Theorems \ref{thm:codedextunique} and \ref{thm:codedextexist}, it remains only to show that if $L$ and $M$ are SFML's, with distinguished central subgroups $Z_L$ and $Z_M$, and associated CVS's $C_L$ and $C_M$, and there is an isomorphism $\phi: L \rightarrow M$ such that $\phi(Z_L)=Z_M$, then $C_L$ and $C_M$ are isomorphic up to scalar multiple. However, since $\sigma$, $\chi$, and $\alpha$ are only determined by the isomorphism type of an SFML, by choosing corresponding generators of $Z_L$ and $Z_M$ (i.e., by applying a scalar multiple), we can make $C_L$ and $C_M$ isomorphic. The theorem follows. \end{proof} \section{Isotopy in small Frattini Moufang 3-loops} \label{sect:isotopy} In this section, as a fairly straightforward application of Main Theorem \ref{mainthm:CVSisSFML}, we characterize isotopy in SFML's. Now, because of Corollary \ref{cor:classtwoGloop}, any SFM $p$-loop is a $G$-loop, unless $p=3$. Therefore, even though much of what we say will apply in general, throughout this section, we assume that $L$ is a finite SFM 3-loop, that $Z$ is a fixed central subgroup of $L$, and that $C\isom L/Z$ is an elementary abelian 3-group. In other words, we assume that $L$ is a coded extension of a CVS $(C,\sigma,\chi,\alpha)$ over $\F_3$. We also resume the convention that $\c,\d,\e,\kappa\in L$ reduce to $c,d,e,k\in C$. \begin{defn} Let $L$ be a Moufang loop. For $\c,\d,\kappa\in L$, we define \begin{equation} \c \circ_\kappa \d = (\c\kappa)(\kappa\i\d). \end{equation} It is easily verified that the set $L$ and the operation $\circ_\kappa$ form a loop $(L,\circ_\kappa)$, with identity 1. Because $(U,V,\iota)$ is an isotopism from $L$ to $(L,\circ_\kappa)$, where $U(x)=x\kappa\i$, $V(x)=\kappa x$, and $\iota(x)=x$, $(L,\circ_\kappa)$ is called the {\it $\kappa$-isotope\/} of $L$. \end{defn} \begin{rem}\label{rem:circinv} Note that the di-associativity of $L$ implies that for $\c\in L$, $\c\i$ is still the inverse of $\c$ in $(L,\circ_\kappa)$. \end{rem} We can now state the following fundamental result on isotopy in Moufang loops (see Pflugfelder~\cite[Thm.\ IV.4.1]{hop:loops}). \begin{thm}\label{thm:isoiskappa} If $L$ is a Moufang loop, then any loop-isotope of $L$ is isomorphic to a $\kappa$-isotope of $L$. \qed \end{thm} The following formula therefore reduces isotopy in SFML's to a matter of multilinear algebra. \begin{thm}\label{thm:kappaformula} We have \begin{equation}\label{eq:kappa} \c \circ_\kappa \d = \c\d\cdot\alpha(c,k,d). \end{equation} \end{thm} \begin{proof} Applying the multilinear and symplectic properties of $\alpha$, we have: \begin{equation} \begin{split} \c \circ_\kappa \d &= (\c\kappa)(\kappa\i\d) \\ &= \c(\kappa(\kappa\i\d)) \cdot\alpha(c,k,k\i d) \\ &= \c\d \cdot\alpha(c,k,d). \qed \end{split} \end{equation} \noqed\end{proof} We now only need the following definitions to proceed. \begin{defn} The {\it radical\/} of $\alpha$ (resp.\ $\chi$), denoted by $\rad(\alpha)$ (resp.\ $\rad(\chi)$), is the set of all $c\in C$ such that $\alpha(c,d,e)=1$ (resp.\ $\chi(c,d)=1$) for all $d,e\in C$ (resp.\ $d\in C$). Note that $\rad(\alpha) = N(L)/Z$ and $\rad(\chi) = C(L)/Z$. \end{defn} \begin{defn} Let $(C,\sigma(c),\chi(c,d),\alpha(c,d,e))$ be a CVS. For $k\in C$, the {\it adjoint translate\/} $\adt_k(C)$ is defined to be the CVS $(C,\sigma(c),\chi(c,d)\alpha(c,k,d),\alpha(c,d,e))$. The operation $\adt_k$ is called an {\it adjoint translation}. \end{defn} %%\begin{rem} %%Note that for $k,m\in C$, multlinearity of $\alpha$ implies that %%\begin{equation} %%\adt_{km}(C) = \adt_k(\adt_m(C)). %%\end{equation} %%In other words, the adjoint translations on a CVS %%$(C,\sigma,\chi,\alpha)$ form a group naturally isomorphic to %%$C/\rad(\alpha)$. %%\end{rem} In the following, we write $\c\circ_\kappa\d$ as $\c\circ\d$. Also, for $\c,\d,\e\in L$, we define $\c^{\circ n}$ inductively by $\c^{\circ 0}=1$ and $\c^{\circ (n+1)} = \c\circ\c^{\circ n}$; we define $[\c,\d]_\circ$ to be $(\d\circ\c)\i(\c\circ\d)$; and we define $[\c,\d,\e]_\circ$ to be $(\c\circ(\d\circ\e))\i((\c\circ\d)\circ\e)$. (See Remark \ref{rem:circinv}.) Main Theorem \ref{mainthm:isotopy} then comes from the following result. \begin{thm}\label{thm:translate} We have the formulas \begin{align} \c^{\circ n} &= \c^n, \label{eq:transsigma} \\ [\c,\d]_\circ &= \chi(c,d)\alpha(c,k,d)\i, \label{eq:transchi} \\ [\c,\d,\e]_\circ &= \alpha(c,d,e). \label{eq:transalpha} \end{align} \end{thm} In particular, $\c^{\circ 3} = \sigma(\c)$. \begin{proof} We first verify \eqref{eq:transsigma} by induction. Applying \eqref{eq:kappa}, we have \begin{equation} \c^{\circ(n+1)} = \c\circ\c^n = \c\c^n\cdot\alpha(c,k,c^n) = \c^{n+1}, \end{equation} with the last equality following from the symplectic property of $\alpha$. \eqref{eq:transsigma} follows. Next, from \eqref{eq:kappa} and \begin{equation} \d \circ_\kappa \c = \d\c\cdot\alpha(d,k,c), \end{equation} it follows that \begin{equation} \begin{split} [\c,\d]_\circ &= (\d\circ\c)\i(\c\circ\d) \\ &= (\d\c)\i(\c\d) \cdot \alpha(d,k,c)\i\alpha(c,k,d) \\ &= [\c,\d] \cdot \alpha(c,k,d)^2 \\ &= \chi(c,d)\alpha(c,k,d)\i, \end{split} \end{equation} with the last equality following because $Z$ has exponent 3. \eqref{eq:transchi} follows. Finally, since \begin{align} (\c\circ\d)\circ\e &= (\c\d)\e \cdot \alpha(c,k,d) \alpha(cd,k,e) \\ \c\circ(\d\circ\e) &= \c(\d\e) \cdot \alpha(d,k,e) \alpha(c,k,de), \end{align} we have that \begin{equation} \begin{split} [\c,\d,\e]_\circ &= (\c\circ(\d\circ\e))\i((\c\circ\d)\circ\e) \\ &= [\c,\d,\e]\cdot\alpha(c,k,d)\alpha(cd,k,e) \alpha(d,k,e)\i\alpha(c,k,de)\i \\ &= \alpha(c,d,e)\alpha(c,k,d)\alpha(c,k,e)\alpha(d,k,e) \\ &\qquad\cdot \alpha(d,k,e)\i\alpha(c,k,d)\i\alpha(c,k,e)\i \\ &= \alpha(c,d,e). \end{split} \end{equation} The theorem follows. \end{proof} \begin{proof}[{\bf Proof of Main Theorem \ref{mainthm:isotopy}}] From \eqref{eq:transsigma}--\eqref{eq:transalpha}, it follows that $(L,\circ_\kappa)$ is the coded extension of $\adt_{k\i}(C)$, since condition \ref{cond:centralext} of Definition \ref{defn:codedext} follows from \eqref{eq:transchi} and \eqref{eq:transalpha}, and the formulas \eqref{eq:CEpower}--\eqref{eq:CEassociate} of condition \ref{cond:formulas} of Definition \ref{defn:codedext} follow from the formulas \eqref{eq:transsigma}--\eqref{eq:transalpha}, respectively. Therefore, any $\kappa$-isotope of $L$ is a coded extension of an adjoint translate of $C$, and vice versa. The theorem follows. \end{proof} We illustrate Main Theorems \ref{mainthm:CVSisSFML} and \ref{mainthm:isotopy} by enumerating the isomorphism and isotopy classes of the nonassociative coded extensions of the CVS's of dimension 3 and 4 over $\F_3$. For simplicity, we only discuss the exponent 3 ($\sigma=1$) cases. In the following, we let $(C,1,\chi,\alpha)$ be the CVS under discussion, we let $L$ be its coded extension, and we let $Z=\gen{\omega}$. \begin{exmp} In the dimension 3 case, either $\chi=1$ (i.e., $L$ is commutative), or there exists a basis $\set{k,c,d}$ of $C$ such that $\rad(\chi)=\gen{k}$ and $\chi(c,d)=\omega$. After possibly inverting $k$, we may assume that $\alpha(c,k,d)=\omega$, so there are two possibilities for $L$, up to isomorphism. However, in the non-commutative case, $\adt_k(C)$ has $\chi=1$ as its bilinear form, so the two isomorphism classes are isotopic. \end{exmp} \begin{exmp} In dimension 4, since we assume $\alpha$ is nontrivial, there is some $c\in C$ such that $c\notin\rad(\alpha)$. Therefore, since $\alpha(c,-,-)$ is a nontrivial bilinear symplectic form on $C$ whose radical contains $c$, we may choose a basis $\set{c,d,e,f}$ for $C$ such that $\alpha(c,d,-)=1$ and $\alpha(c,e,f)=\alpha(d,e,f)\i$. It may then be easily verified on the basis $\set{c,d,e,f}$ that $cd\in\rad(\alpha)$. Therefore, $\rad(\alpha)$ is 1-dimensional. It follows easily that $\alpha$ is unique up to isomorphism, and that we have four possible isomorphism classes for $C$ (and therefore, for $L$): \begin{enumerate} \item\label{enum:chitriv} $\chi$ trivial; \item\label{enum:chinon} $\chi$ nondegenerate; \item\label{enum:radin} $\rad(\chi)$ 2-dimensional, containing $\rad(\alpha)$; and \item\label{enum:radnot} $\rad(\chi)$ 2-dimensional, not containing $\rad(\alpha)$. \end{enumerate} So now let $\rad(\alpha)=\gen{c}$. First, for $d\in C$, $\chi(c,d)$ is invariant under adjoint translation, so isomorphism classes \ref{enum:chitriv} and \ref{enum:chinon} cannot be adjoint translates. Conversely, if $\chi$ is nondegenerate, without loss of generality, we may choose a basis $\set{c,k,d,e}$ for $C$ such that $\gen{c,k}$ and $\gen{d,e}$ are orthogonal with respect to $\chi$ and $\chi(d,e)=\alpha(d,k,e)$, in which case $\adt_k(C)$ is in isomorphism class \ref{enum:radnot}. On the other hand, if $\rad(\alpha)=\gen{c}$ and $\rad(\chi)=\gen{c,k}$, then by inverting $k$ if necessary, we may choose a basis $\set{c,k,d,e}$ for $C$ such that $\chi(d,e)=\alpha(d,k,e)$, in which case $\adt_k(C)$ is in isomorphism class \ref{enum:chitriv}. Therefore, we have precisely two isotopy classes: isomorphism classes \ref{enum:chitriv} and \ref{enum:radin}, and isomorphism classes \ref{enum:chinon} and \ref{enum:radnot}. \end{exmp} Compare B\'{e}n\'{e}teau~\cite[IV.3]{lb:cml} and Ray-Chaudhuri and Roth~\cite{dkrcrr:hallcml}. \section{A construction of all finite Moufang loops of class 2} \label{sect:constructclasstwo} In this section, we give a construction of all finite Moufang $p$-loops of class 2, which, because of Theorem \ref{thm:torsionsubloop}, gives a construction of all finite Moufang loops of class 2. We will mostly be imitating Sections \ref{sect:eml}--\ref{sect:isotopy}, using modules instead of vector spaces, so many details will be omittted. \begin{notn} For the rest of this section, we let $p$ be a prime, we let $L$ be a Moufang $p$-loop with a central subgroup $Z$ such that $C\isom L/Z$ is an abelian group, and we resume our $\c,\d,\e,\dots$ and $c,d,e,\dots$ convention. \end{notn} We would like to say that $L$ gives $C$ the structure of a ``coded module,'' whatever that means. Now, $\chi$ and $\alpha$ work exactly as they do for SFML's (Theorem \ref{thm:sympmulti}). However, if $r$ is the exponent of $Z$, the only well-defined ``power functions'' $\sigma: C \rightarrow Z$ are the $q$th power functions, where $r$ divides $q$, so if $r>p$, these power functions no longer capture all of the information we need. We fix this problem with the following definition. \begin{defn}\label{defn:codmod} A {\it coded module\/} is defined to be a 6-tuple $(C,Z,\set{c_i},\set{z_i},\chi,\alpha)$, where $C$ and $Z$ are finite abelian $p$-groups; $\set{c_i}$ is a basis (set of independent generators) for $C$; $\set{z_i}$ is a set of elements of $Z$; and $\chi: C\times C\rightarrow Z$ and $\alpha: C\times C\times C\rightarrow Z$ satisfy \eqref{eq:chisymp}--\eqref{eq:chimultilin} and \eqref{eq:alphasymp}--\eqref{eq:alphamultilin} for all $c,d,e,f\in C$ and all $n\in\Z$. \end{defn} We then see that $L$, along with a choice of basis $\set{c_i}$ for $C$ and a choice of preimages $\c_i$ in $L$, makes $C$ and $Z$ into a coded module, by defining $z_i = \c_i^{q_i}$, where $q_i$ is the order of $c_i$, and defining $\chi$ and $\alpha$ as usual. We may now recover much of Sections \ref{sect:eml} and \ref{sect:codedextensions} in the case of coded modules. First, we can generalize the definition of coded extension (Definition \ref{defn:codedext}) to coded modules by removing the condition that $Z$ have order $p$, and replacing \eqref{eq:CEpower} with \begin{equation}\label{eq:codedmodpower} \c_i^{q_i} = z_i. \end{equation} Imitating Theorem \ref{thm:codedextisSFML}, we see that a loop $L$ (with choice of preimages, etc.) is a coded extension of $C$ if and only if $L$ is a Moufang $p$-loop of class 2 whose coded module is $C$. Having carried over the basic definitions, we next generalize the results of Section \ref{sect:codedextensions} appropriately. First, the uniqueness of coded extensions follows from the proof of Theorem \ref{thm:codedextunique}, replacing \eqref{eq:Cpowerreln} with \eqref{eq:codedmodpower}, and \eqref{eq:zpower} with defining (group) relations for $Z$. The definition of the semidirect central product (Definition \ref{defn:sdcp}) also still works, as does Theorem \ref{thm:sdcpiscodedext}, if we replace the verification of \eqref{eq:CEpower} by the observation that concatenation of the bases of $D$ and $E$ produces a basis for $C$. Finally, coded extensions of 1-dimensional coded modules are easy to construct using central products of groups, so Theorem \ref{thm:codedextexist} follows as before. In fact, in general, the only result not carried over from Section~\ref{sect:codedextensions} is the ``isomorphic SFML's implies isomorphic CVS's'' statement of Main Theorem \ref{mainthm:CVSisSFML}, since the structure of a coded module is highly basis-dependent. However, if $Z$ is elementary abelian, then by defining $\sigma_q(c): C_q\rightarrow Z$ by $\sigma_q(c)=\gamma^q$, for all $q=p^n$, where $C_q$ is the subgroup of $C$ of all elements of $C$ whose order divides $q$, we may actually recover all of Main Theorem \ref{mainthm:CVSisSFML}. In fact, since \eqref{eq:sigmalin} holds in the elementary abelian case if we replace $p$ with $q$, everything works as before. (In particular, when $q$ is a power of 2, $\sigma_q$ is linear for $q>2$.) We can also generalize Main Theorem \ref{mainthm:isotopy} to the case where $Z$ is an elementary abelian 3-group and $C$ is a coded module with values in $Z$. We state this result in the following theorem, whose proof may be copied directly from Section \ref{sect:isotopy}. (Note that the proof of Theorem \ref{thm:translate} only uses the exponent of $Z$, and not its order.) \begin{thm}\label{thm:expthreeisotopy} Let $L$ be a coded extension of a coded module $C$ of 3-power order by an elementary abelian 3-group. Then up to isomorphism, the loop-isotopes of $L$ are precisely the coded extensions of the adjoint translates of $C$. In particular, $\sigma_q$ ($q=3^n$) and $\alpha$ are ``isotopy invariants'' of $L$. \qed \end{thm} For example, Theorem \ref{thm:expthreeisotopy} applies when $L$ has exponent 3 and class 2. Finally, we address the question analogous to the one posed in Section \ref{sect:cvsiscode}: that of which $\chi$, $\alpha$, etc., can be chosen for a coded module. (The 1-dimensional case shows that any values may be chosen for the $z_i$.) Again, for $p>2$, this is easy (keeping in mind that $\alpha^6=1$), so it will be enough to prove Theorem \ref{thm:consistenttwo}, in which we use the following conventions. \begin{notn} We revert to additive notation for abelian groups. All summations, indices, etc., are from 1 to $n$ in the manner indicated. (For instance, $ij} \sum_k c_i d_j e_k \alpha_{ijk} \\ {}={} &\alpha(c,d,e), \end{split} \end{equation} using $\alpha_{ijk}=\alpha_{jik}$ and $\alpha_{kjk}=\alpha_{ikk}=0$. The theorem follows. \end{proof} Finally, when $Z$ is an elementary abelian $p$-group, we have seen previously that it is possible to define $\sigma_q: C_q\rightarrow Z$ for all $q=p^n$, and so the problem arises of which values can be chosen for $\sigma_q$. Now, since $\sigma_q$ is trivial on $C_{q/p}$, for $q>2$, we may choose $\sigma_q$ freely on a basis for $C_q/C_{q/p}$, since $\sigma_q$ is linear for $q>2$. Therefore, the only case remaining is $\sigma=\sigma_2$. We leave it to the interested reader to verify (by imitating the proof of Theorem \ref{thm:consistenttwo}) that if we choose arbitrary elements $\sigma_i\in Z$, and define \begin{equation}\label{eq:sigmatwo} \sigma(c) = \sum_i c_i \sigma_i + \sum_{i