JOURNAL OF ALGEBRA 184, 213]229Ž. 1996 ARTICLE NO. 0257 Loops with Transitive Automorphisms* Arthur A. Drisko² Department of Mathematical Sciences, Uni¨ersity of Texas at El Paso, El Paso, Texas 79968-0514 Communicated by Walter Feit CORE Metadata, citation and similar papers at core.ac.uk Received November 6, 1995 Provided by Elsevier - Publisher Connector It is shown that isotopic loops with transitive automorphism groups are in fact isomorphic. A classification of loops with transitive automorphism groups is given. This classification is compared to one given by Barlotti and StrambachŽ. 1983 for loops with sharply transitive automorphism groups, and examples of several of the classes are presented. The approach is entirely algebraic. Q 1996 Academic Press, Inc. 1. INTRODUCTION Several properties P of loops are known such that isotopic loops with property P are isomorphicwx 5, pp. 57, 58 . For example, isotopic free loops are isomorphicwx 7 , isotopic commutative Moufang loops are isomorphic, and isotopic totally symmetric loops are isomorphicwx 3 . Here we shall prove that if two loops are isotopic and both have transitive automorphism groups, then they are in fact isomorphic. This result generalizes an earlier theorem of the author showing that this is the case when the loops in n question have order p q 1, where p is a primewx 6 . AquasigroupŽ. G, ? is a set G with a binary operation ? such that for all y, z g G each of the equations x ? y s z y ? x s z * Part of this work was done while the author was supported by NSF Grant DMS-9303379 at the University of California at Berkeley. ² E-mail: [email protected]. 213 0021-8693r96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. 214 ARTHUR A. DRISKO has a unique solution for x. G is a loop if it is a quasigroup and has a two-sided identity element eG. For details on quasigroups and loops, see wx5or15. w x An isotopism Ž.a, b, g from a quasigroup Ž.G, ? to a quasigroup Ž.H, ( is a triple of bijections a, b, g from G to H such that for all x, y, z g G, x ? y s z m xa ( yb s zg .1Ž. An autotopism is an isotopism from a quasigroup to itself. The set of autotopisms of a quasigroup G forms a group AŽ.G .Anisomorphism is an isotopism such that a s b s g. In this case we shall sometimes denote the isomorphism by the single map a. Note that an isomorphism of loops must take the identity of one loop to the identity of the other. The automor- phism group, AutŽ.G , of a quasigroup G is clearly a subgroup of A Ž.G . The automorphism group of a loop is said to be transiti¨e if it acts transitively on the set of nonidentity elements of G. We shall sometimes abbreviate ``transitive automorphism group'' to ``tag,'' saying that a loop with transitive automorphism group has tag and calling it a tag-loop. Bruck studied tag-loopswx 4, 5 , showing, for example, that finite tag-loops are simple. Examples of tag-loops include neofields, introduced by Paige wx14 ; one-sided neofields w 12, 9 x ; and division neo-rings, introduced by Hugheswx 10 . In Section 2 we present someŽ. standard loop-theoretic lemmas and develop a criterion for isomorphism of isotopic loops. In Section 3 we specialize to tag-loops and prove the main theorem, that whenever two such loops are isotopic, they are isomorphic. Finally, in Section 4 we give a classification of tag-loops and examples of each class, and compare the results with the classification of loops with sharply transitive automor- phism groups given by Barlotti and Strambach inwx 1 . 2. LOOP-THEORETIC PRELIMINARIES In this section we present the loop-theoretic results which we shall need in what follows. Most of these can be found, either explicitly or implicitly, in the literatureŽ see especiallywx 15. , but we include them in order to set up a framework for the results in the following sections. When considering the action of isotopisms and autotopisms of loops, we shall often find it convenient to think of a loop Ž.G, ? in terms of the set Ž. TG of all ordered triples x, y, z of elements of G such that x ? y s z. The definition of a loop guarantees that any two coordinates of such a triple uniquely determine the third. The set TG may be regarded as the Cayley table of G, where the first coordinate denotes row, the second denotes LOOPS WITH TRANSITIVE AUTOMORPHISMS 215 column, and the third denotes entry. An isotopism Ž.a, b, g from G to H Ž. Ž . takes x, y, z g TGHto xa, yb, zg g T . Hence, by an abuse of nota- tion, we shall write GŽ.Ža, b, g s H. This point of view is closely related to the concept of a net or web wx1, 15 , but we shall not make explicit use of nets in this paper.. Ž. Ž. Given a loop G, ? and an element x g G, let RxG :GªGdenote Ž. right multiplication by x; i.e., for all y g G, yRG x s y ? x. Similarly LxGŽ.will denote left multiplication by x. When it is clear in which loop the multiplication is taking place, we shall drop the subscript and write RxŽ.or Lx Ž.. Our first lemma characterizes certain special isotopisms which we shall encounter frequently. LEMMA 1. Let Ž.a, b, g be an isotopism from the loop Ž. G, ? to the loop Ž.H,(. Ž. Ž . Ž . a bLeHGasgsaReHGb. Ž. Ž . Ž Ž . Ž .. b If eGHa s e , then a, b, g s a, aReHGHGb,aReb. Ž. Ž . Ž Ž . Ž .. c If eGHb s e , then a, b, g s bLeHGHGa,b,bLea. Ž. Ž. d If two of a, b, g take eGH to e , then a s b s g, so a, b, g is an isomorphism. Ž. Proof. a For all x g G we have eG? x s x,so xgseGa(xb, Ž. whereby we see that g s bLeHGa. On the other hand, x ? eGs x gives us xg s xa ( eG b , Ž. Ž. Ž. Ž. so g s aReHGb. Statements b and c follow immediately, and d follows easily fromŽ. b and Ž. c . Let G be a loop. A bijection a: G ª G is a right pseudo-automorphism of G if there exists c g G such that ŽŽ.Ž..Ž.a, aRc,aRc gAG. In this case, c is called a companion of a. Note that applying ŽŽ.Ž..a, aRc,aRc to e ? e s e gives us ea s e. From this and partŽ. b of the lemma, it is easy to show that the set of right pseudo-automorphisms of G forms a group Ž. Pr G, consisting of all permutations a of G such that ea s e and Ž.Ž.a,b,ggAGfor some b, g. Left pseudo-automorphisms are defined analogously and form a group PlŽ.G . Unfortunately, the term ``middle pseudo-automorphism'' does not seem to exist in the literature. Let us define it now. Note that if Ž.Ž.a, b, g g A G , then y1 y1 Ž.Ž.Ž.a , b , g s Ž.g Reb ,gLea ,g, 216 ARTHUR A. DRISKO by partŽ. a of the lemma. If, furthermore, eg s e, then ea ? eb s e. 1 1 Conversely, let ŽŽ.g Ryy,gLxŽ.y,g.Ž.gAG and x ? y s e. Applying 1 1 1 ŽŽ.gRyy,gLxŽ.y,g.y to x ? y s e gives us 1 1 1 egy s xRŽ. y gy ? yL Ž. x gy 1 1 sŽ.x?ygy ? Ž.x?ygy 1 1 segy?egy, 1 but since the only idempotent in a loop is the identity, egy s e,so egse. In this case we shall call g a middle pseudo-automorphism withŽ. right companion y. The set of all middle pseudo-automorphisms forms a group PmŽ.Gas before. Throughout this section and the next, we shall assume that whenever any of the loops we are considering have the same cardinalityŽ in particu- lar, if they are isotopic. , they have the same underlying set, denoted by X. For example, if G and H are finite loops of order n, we may assume that GsŽ.X,?,Hs Ž.X,(, with X s Ä41,...,n. This convention allows us to view isotopisms between loops as triples of permutations of X. We shall use i to denote the identity permutation of X. Given a loop G s Ž.X, ? with identity e and elements f, g g G, we can Ž. define a new loop Gf, g s X, ( by y1 y1 x( y s xRGGŽ. g ? yL Ž. f .2Ž. In other words Gf,gGGs GRŽ.Ž. g,Lf Ž.,i.3Ž. Gf, g has identity element f ? g and is called the principal f, g-isotope of G. LEMMA 2. Assume that the loop H is isotopic to the loop G. Then H is isomorphic to Gf, g for some f, g g G. 1 1 Proof. Let Ž.Ž.Ž.H,) s G, ? a, b, g , d s agy, e s bgy,and Ž.Ž.Ž.K,(sG,?d,e,i. We have the following commutative diagram: Ž.a,b,g6 GH6 , Ž.d,e,i6 Ž.g,g,g K LOOPS WITH TRANSITIVE AUTOMORPHISMS 217 Ž.