CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 130, 385-387 (1990) Code Loops Are R42 Loops ORIN CHEIN * Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122 AND EDGAR G. GOODAIRE~ Department of Mathematics and Statisrics, Memorial University of Newfoundland, St. John’s, Newfoundland, Canada AIC 5S7 Communicated by George Glauberman Received December 22, 1988 1. INTRODUCTION In [9], Griess considers a class of loops, called code loops, which have applicability to a construction of the monster and its nonassociative algebra (see [7]). In [6], the authors characterize these loops as Moufang loops which have at most one nonidentity square (and hence also at most one nonidentity commutator and at most one nonidentity associator). Moufang loops with one or more of these “uniqueness” properties play an important role in the authors’ work on loops which have alternative loop rings [2-5, 81. For example, a nonassociative loop which has an alter- native loop ring over a ring of characteristic different from two must be a Moufang loop with a unique nonidentity commutator and associator (which coincide). It is the purpose of the present paper to prove that a nonassociative Moufang loop with a unique nonidentity square (i.e., a nonassociative code loop) must, in fact, have an alternative loop ring over any ring of characteristic two. * The research of this author was supported by a grant from the American Philosophical Society and by a research leave from Temple University. + The research of this author was supported in part by Grant A9087 from the Natural Sciences and Research Council of Canada. 385 OO21-8693/90 $3.00 Copyright 0 1990 by Academic Press, Inc. All rights of reproduction in any form reserved. 386 CHEIN AND GOODAIRE 2. BACKGROUND AND NOTATION We assume that the reader has some familiarity with the terminology of Moufang loops. For those readers who do not, [6] contains all the necessary information. (See [l] for a full treatment of the subject.) The commutator subloop, associator subloop, and center of a Moufang loop L will respectively be denoted by L’, A(L), and Z(L); and L2 will denote the subloop generated by the squares of all elements in L. An RA2 loop is a nonassociative loop whose loop ring over some ring of characteristic two is alternative. Theorem 2.1 of [S] characterizes RA2 loops as Moufang loops in which each three elements x, y, z either associative or else satisfy one of the following pairs of equations: xy.z=y.xz and x.yz=yx.z (1) or xy.z= yx.z and x.yz=y.xz. (2) (Note that the first equation in (2) can be written more simply as xy = yx; but it has been written as above to emphasize the fact that the second equations in (1) and (2) are obtained from the first by shifting the multi- plication dot.) Since this characterization does not depend in any way on the ring, the property of being an RA2 loop depends only on the loop itself and not on the particular ring. That is, if L has an alternative loop ring over one ring of characteristic two, then it has alternative loop rings over all such rings. We also will need Theorem 1 of [6]: If L is a nonassociative Moufang loop with a unique nonidentity square e, then e2 = 1 and L’ = A(L) = L2= (l,e} EZ(L). 3. THE MAIN RESULTS We are now ready to state and prove the main results of this paper: THEOREM. If L is a nonassociative Moufang loop with a unique non- identity square, e, then L is an RA2 loop. COROLLARY. Every nonassociative code loop L is an RA2 loop. ProoJ From the remarks in the Introduction (see also Theorem 5 of [6]), every code loop contains at most one nonidentity square. If all squares are the identity element, then so are all commutators, since (x, y) =x-‘y-‘xy =x-~(xY-‘)~ y2, and so L is a commutative Moufang CODE LOOPS ARE RA2 LOOPS 387 loop. But then, by Lemma VII.5.7 of [ 11, x3 E Z(L) for all x in L. But x2 = 1 so x3 =x and thus every element of L is central; that is, L is an abelian group, contrary to assumption. Thus every nonassociative code loop must contain a nonidentity square, and the Corollary follows from the Theorem. 1 Proof of Theorem. By Theorem 1 of [6], every non-trivial commutator and every non-trivial associator of elements of L must equal e. Let x, y, z be any three elements of L which do not associate. We consider two cases (using the centrality of e and the fact that e2 = 1 without further comment). Case 1. If xy=yx, then (xy)z=(yx)z and also x(yz)=[(xy)z]e= [ (yx)z] e = y(xz). Hence (2) holds. Case 2. If xy#yx, then x(yz)= [(xy)z]e= [(yx.e)z]e= (yx)z, and so (xy)z= [x(yz)]e= [(yx)z]e=y(xz). Hence (1) holds. Thus, for any x, y, z in L, either (x, y, z) = 1 or else (1) or (2) holds, and so L is RA2, by Theorem 2.1 of [S]. 1 Remark. By Theorem 2.1 of [5], if IL21=2, then IL’\ = IA(L)\ = 2. In Theorem 3.2 of [S], we prove a weak converse of our current Theorem: If L is an RA2 loop with a noncommutative nucleus, then IL’1 = 1A( = 2. ACKNOWLEDGMENTS The authors thank the Department of Mathematics of the University of British Columbia for its hospitality during the period when the research was accomplished. REFERENCES 1. R. H. BRUCK, A survey of binary systems, in “Ergeb. Math. Grenzgeb.,” Vol. 30, Springer- Verlag, Berlin, 1958. 2. 0. CHEIN AND E. G. GOODAIRE, Isomorphism of loops which have alternative loop rings, Comm. Algebra 13 (1985), l-20. 3. 0. CHEIN AND E. G. GOODAIRE, Loops whose loop rings are alternative, Comm. Algebra 14 (1986), 293-310. 4. 0. CHEIN AND E. G. GOODAIRE, Moufang loops with limited commutativity and one commutator, Arch. Math. 51 (1988), 92-96. 5. 0. CHEIN AND E. G. GCIODAIRE,Loops whose loop rings over rings of characteristic two are alternative, Comm. Algebra, in press. 6. 0. CHEIN AND E. G. GOODAIRE, Moufang loops with a unique nonidentity commutator (associator, square), J. Algebra 130 (1990), 369-384. 7. J. CONWAY, A simple construction of the Fischer-Griess monster group, Invent. Math. 79 (1985), 513-540. 8. E. G. GOODAN, Alternative loop rings, Publ. Math. Debrecen 30 (1983), 31-38. 9. R. L. GRIEFS, JR., Code loops, J. Algebra 100 (1986), 224234. 481/130/2-9 .