JOURNAL OF ALGEBRA 184, 213᎐229Ž. 1996 ARTICLE NO. 0257
Loops with Transitive Automorphisms*
Arthur A. Drisko†
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. ᮊ 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, и is a set G with a binary operation и such that for all y, z g G each of the equations
x и y s z y и 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. † E-mail: [email protected].
213
0021-8693r96 $18.00 Copyright ᮊ 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.␣, , ␥ from a quasigroup Ž.G, и to a quasigroup Ž.H, ( is a triple of bijections ␣, , ␥ from G to H such that for all x, y, z g G,
x и y s z m x␣ ( y s z␥ .1Ž.
An autotopism is an isotopism from a quasigroup to itself. The set of autotopisms of a quasigroup G forms a group ᑛŽ.G .Anisomorphism is an isotopism such that ␣ s  s ␥. In this case we shall sometimes denote the isomorphism by the single map ␣. 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 ᑛ 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, и in terms of the set Ž. TG of all ordered triples x, y, z of elements of G such that x и 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.␣, , ␥ from G to H Ž. Ž . takes x, y, z g TGHto x␣, y, z␥ g T . Hence, by an abuse of nota- tion, we shall write GŽ.Ž␣, , ␥ 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, и 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 и 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.␣, , ␥ be an isotopism from the loop Ž. G, и to the loop Ž.H,(. Ž. Ž . Ž . a LeHG␣s␥s␣ReHG. Ž. Ž . Ž Ž . Ž .. b If eGH␣ s e , then ␣, , ␥ s ␣, ␣ReHGHG,␣Re. Ž. Ž . Ž Ž . Ž .. c If eGH s e , then ␣, , ␥ s LeHGHG␣,,Le␣. Ž. Ž. d If two of ␣, , ␥ take eGH to e , then ␣ s  s ␥, so ␣, , ␥ is an isomorphism. Ž. Proof. a For all x g G we have eGи x s x,so
x␥seG␣(x, Ž. whereby we see that ␥ s LeHG␣. On the other hand, x и eGs x gives us
x␥ s x␣ ( eG  , Ž. Ž. Ž. Ž. so ␥ s ␣ReHG. Statements b and c follow immediately, and d follows easily fromŽ. b and Ž. c .
Let G be a loop. A bijection ␣: G ª G is a right pseudo-automorphism of G if there exists c g G such that ŽŽ.Ž..Ž.␣, ␣Rc,␣Rc gᑛG. In this case, c is called a companion of ␣. Note that applying ŽŽ.Ž..␣, ␣Rc,␣Rc to e и e s e gives us e␣ 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. ⌸ G, consisting of all permutations ␣ of G such that e␣ s e and Ž.Ž.␣,,␥gᑛGfor some , ␥. Left pseudo-automorphisms are defined analogously and form a group ⌸Ž.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.␣, , ␥ g ᑛ G , then
y1 y1 Ž.Ž.Ž.␣ ,  , ␥ s Ž.␥ Re ,␥Le␣ ,␥, 216 ARTHUR A. DRISKO by partŽ. a of the lemma. If, furthermore, e␥ s e, then e␣ и e s e. 1 1 Conversely, let ŽŽ.␥ Ryy,␥LxŽ.y,␥.Ž.gᑛG and x и y s e. Applying 1 1 1 ŽŽ.␥Ryy,␥LxŽ.y,␥.y to x и y s e gives us
1 1 1 e␥y s xRŽ. y ␥y и yL Ž. x ␥y 1 1 sŽ.xиy␥y и Ž.xиy␥y 1 1 se␥yиe␥y,
1 but since the only idempotent in a loop is the identity, e␥y s e,so e␥se. In this case we shall call ␥ a middle pseudo-automorphism withŽ. right companion y. The set of all middle pseudo-automorphisms forms a group
⌸Ž.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,и,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 to denote the identity permutation of X. Given a loop G s Ž.X, и 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 и yL Ž. f .2Ž.
In other words
Gf,gGGs GRŽ.Ž. g,Lf Ž.,.3Ž.
Gf, g has identity element f и 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, и ␣, , ␥ , ␦ s ␣␥y, ⑀ s ␥y,and Ž.Ž.Ž.K,(sG,и␦,⑀,. We have the following commutative diagram:
Ž.␣,,␥6 GH6
, Ž.␦,⑀,6 Ž.␥,␥,␥ K LOOPS WITH TRANSITIVE AUTOMORPHISMS 217
Ž. y1 Now ␥, ␥, ␥ is an isomorphism, so K is a loop. Let f s eK ␦ and y1 g s eK⑀ . Now x␦ ( y⑀ s x и y,so
x␦sx␦(eyKK⑀se(y⑀ sx␦(g⑀sf␦(y⑀ and s x и g s f и y
s xRGGŽ. g s yLŽ. f , Ž. Ž. Ž. so ␦ s RgGGand ⑀ s Lf. Hence by 3 we have
KsGRŽ.GGŽ. g,Lf Ž.,sG f,g, and H is isomorphic to K. Lemma 2 shows that to prove any fact about loops isotopic to a given loop G, one need only examine the principal f, g-isotopes of G. In order to prove our main result in Section 3, we shall need the following criterion for determining when two of these are isomorphic. Here and throughout the rest of the paper, we shall often use , for isomorphism; xy for x и y, Ž. where G s X, и ; and R and L for RGGand L .
LEMMA 3. Let G be a loop. Then Gf, gc is isomorphic to G,d if and only if there exists Ž.Ž.Ž.Ž.Ž.␣, , ␥ g ᑛ G such that f, g, fg ␣, , ␥ s c, d, cd .
Proof. Assume Gf, gc, G ,d. Then there is some permutation of X such that
Gf,gcŽ. , , s G ,d.4Ž. Ž. Since fg and cd are the identities of Gf, gcand G ,d, respectively, fg s cd. ByŽ.Ž. 3 , 4 is the same as
GRgŽ.Ž.Ž.,Lf Ž., Ž,, .sGRd Ž.,Lc Ž.,. Hence,
y1 y1 Ž.RgŽ.,Lf Ž., Ž,, .Ž.Rd Ž.,Lc Ž., gᑛ ŽG .. It is easy to see that this autotopism takes Ž.Ž.f, g, fg to c, d, cd . Conversely, suppose that Ž.Ž.Ž.Ž.␣, , ␥ g ᑛ G and f, g, fg ␣, , ␥ s Ž.c,d,cd . Then Ž.RgŽ.y1 ,Lf Ž.y1 ,␣ Ž,,␥ .Ž.Rd Ž.,Lc Ž., is an isotopism from Gf, gcto G ,d, and it is easy to see that it takes Ž.Ž.fg, fg, fg to cd, cd, cd . By part Ž. d of Lemma 1, since all three coordi- nates of this isotopism send the identity of Gf,gcto the identity of G ,d,itis an isomorphism. 218 ARTHUR A. DRISKO
This lemma allows us to think of an isomorphism class of loops isotopic to G as an orbit of triples in TG under the action of the autotopism group of G. The following corollary is immediate.
COROLLARY 4. A loop G is isomorphic to all of its loop isotopes if and only if for all f, g g G, there exists Ž.Ž.␣, , ␥ g ᑛ G such that Ž.Ž.Ž.e,e,e␣,,␥sf,g,fg . A loop which is isomorphic to all of its loop isotopes is known as a G-loop. The following corollary is the G-loop criterion which is usually given in the literaturewx 2, 1, 15 . COROLLARY 5. A loop G is a G-loop if and only if e¨ery element of G is a companion of a left and of a right pseudo-automorphism.
Proof. If G is isomorphic to all of its loop isotopes, then for any c g G, there exists Ž.Ž.Ž.Ž.Ž.␣, , ␥ g ᑛ G such that e, e, e ␣, , ␥ s e, c, c .By Lemma 1, ␣ is a right pseudo-automorphism with companion c.Bya similar argument c is also a companion of a left pseudo-automorphism. Conversely, given f, g g G, let ␣ be a right pseudo-automorphism with companion g and let  be a left pseudo-automorphism with companion f␣y1. Then the autotopism Ž.LfŽ.␣y1 ,,Lf Ž.␣y1 Ž.␣,␣RgŽ.,␣Rg Ž. Ž.Ž. takes e, e, e to f, g, fg . Hence G , Gf, g . Note that if G is a group, then it is a G-loop, since for all x g G, Ž, Rx Ž .,Rx Ž ..and ŽLx Ž .,,Lx Ž .. are autotopisms.
3. THE MAIN RESULT
We now turn our attention to loops with transitive automorphism groups, or tag-loops for short. Given any loop G, let
Aiis AGŽ.sÄ4 Ž␣123,␣,␣ .gᑛ Ž.GNe␣ise. Ž. Note that Aijl A s Aut G whenever i / j, by Lemma 1. The subgroups of ᑛŽ.G may be arranged as in Fig. 1, where a line segment denotes inclusion of the lower subgroup in the upper.
In view of Lemma 1, the group A1 is of course related to the group of right pseudo-automorphisms ⌸. In fact, if we let 1 be the projection Ž. map of ᑛ G onto its first coordinates, then A11 s ⌸, and the kernel < Ž of 1 A1 is isomorphic to the right nucleus of G seewx 5 or w 15 x for the definitions of nuclei. . However, we shall not need this fact or the corre- sponding statements for A23and A . LOOPS WITH TRANSITIVE AUTOMORPHISMS 219
FIG. 1. Subgroups of ᑛŽ.G .
LEMMA 6. Let G be any loop. The right cosets of AutŽ.G in A12 Ž resp., A , A3. are in one-to-one correspondence with the elements of G which are companions of rightŽ. resp., left, middle pseudo-automorphisms.
Proof. Every element of A1 is of the form ŽŽ.Ž..␣, ␣Rx,␣Rx ,by Lemma 1. Let Ž␣11, ␣ Rx Ž .,␣ 1Rx Ž ..and Ž␣22, ␣ Ry Ž .,␣ 2Ry Ž .. be two elements of A1. These are in the same right coset of AutŽ.G if and only if
y1 Ž.␣11, ␣ RxŽ.,␣ 1Rx Ž.Ž.␣ 22,␣Ry Ž.,␣ 2Ry Ž.gAut ŽG ..
This holds if and only if all of the coordinates of this product are equal, y1 Ž. Ž.y1y1 namely, ␣␣12s␣ 1RxRy ␣2. This holds if and only if x s y. The analogous statements for A23and A are proved in the same way. COROLLARY 7. Let G be a loop with transiti¨e automorphism group. Then Ž. Ž. for each i s 1, 2, 3, either Aiis Aut G or the right cosets of Aut GinA are in one-to-one correspondence with the elements of G.
Proof. We prove the case i s 1; the others are similar. If e is the only companion of a right pseudo-automorphism, then every element of A1 is an automorphism. If x / e is the companion of some right pseudo-auto- morphism ␣, then ŽŽ.Ž..Ž.␣, ␣Rx,␣Rx gᑛGand e␣ s e. Given y / e in G, choose g AutŽ.G such that x s y. Then e␣Rx Ž.sy, and ŽŽ.Ž.. ␣, ␣Rx,␣Rx gA1,so yis a companion of the right pseudo- automorphism ␣, by Lemma 1. Hence every element of G is such a companion and the statement follows from Lemma 6. The next lemma and its corollary are completely analogous to the above, so we omit the proofs.
LEMMA 8. Let G be any loop. For each i s 1, 2, 3, the right cosets of Ai in ᑛŽ.G are in one-to-one correspondence with the elements of G which are images of e under the i th coordinate of some autotopism. 220 ARTHUR A. DRISKO
COROLLARY 9. Let G be a loop with transiti¨e automorphism group. Then Ž. Ž. for each i s 1, 2, 3, either ᑛ G s Aii or the right cosets of A in ᑛ G are in one-to-one correspondence with the elements of G.
The two corollaries together tell us that for a tag-loop G, each subgroup of ᑛŽ.G in Fig. 1 has index 1 or < Since in this class AutŽ.G has index < THEOREM 10. Any two isotopic loops with transiti¨e automorphism groups are isomorphic. Proof. By Lemma 2, it suffices to show that if G is a tag-loop and Gf, g is a tag-loop, then G , Gf, g . Fix a tag-loop G. We need only show, by FIG. 2. Classes of tag-loops. LOOPS WITH TRANSITIVE AUTOMORPHISMS 221 Lemma 3, that if Gf, g has tag, then G has an autotopism taking Ž.e, e, e to Ž.f,g,fg . If G is in class G, it is isomorphic to all of its loop isotopes, and we are done, so we may assume G is not in class G. We also assume, for the ŽŽ. Ž.. moment, that G is not in class C1. Now Gf,gs GRg,Lf,,so y1 y1 Ž.RgŽ.,Lf Ž.,AutŽ.GRgf,gŽ.Ž.,Lf Ž. , Ž.5 is a subgroup of ᑛŽ.G , whose first coordinates fix f, second coordinates fix g, and third coordinates fix fg. Denote this subgroup by H. Since AutŽ.Gf, g is transitive, there are no other fixed points of H in any coordinate. If G in class A, then H - AutŽ.G , so we must have f s e s g, whereby Gf, ges G ,es G, and we are done. Hence we may assume that G is not in class A. Then there exists Ž.Ž.␣, , ␥ g ᑛ G such that e␣ s f, by Lemma 8 and Corollary 9. Then Ž.␣ ,  , ␥ HŽ.␣y1 , y1 , ␥y1 Ž.6 is a subgroup of ᑛŽ.G . Denote it by K. The first coordinates of K fix e but act transitively on the other elements of G, so we have K - A1. Since Ž. Ž. Gis not in class G or C11, we have A s Aut G ,so K-Aut G , whereby every element of K fixes e in each coordinate. Looking at the second coordinate, we see that y1 y1 eLfŽ.AutŽ.GLff,g Ž. se, so eLfŽ.is a fixed point of AutŽ.Gf, g and is hence equal to fg. Thus esg,soŽ.Ž.Ž.e,e,e ␣,,␥ s f,g,fg . If G is in class C1, we use a similar argument, but choose Ž.␣, , ␥ such that e s g and show that e␣ s f. 4. CLASSIFICATION In this section we give a classification of tag-loops, compare it to Barlotti and Strambach’s classification of loops with sharply transitive automor- phism groupswx 1, Corollary 17.3 , and give examples. Recall that given a loop G, we have the set T s TGs Ä4Ž.x, y, z N x, y, z g G, xy s z . 222 ARTHUR A. DRISKO We define several subsets of T: T0s Ä4Ž.e, e, e T1sÄ4Ž.e,x,xNxgG_Ä4e T2sÄ4Ž.y,e,yNygG_Ä4e T3sÄ4Ž.w,z,eNw,zgG_Ä4e,wz s e T4sT_Ž.T 0123jTjTjT. Under the action of the autotopism group ᑛŽ.G , for each i s 0, 1, 2, 3, the elements of Ti lie in a single orbit, by the transitivity of the automorphism group. Recall from Section 2 that each isomorphism class of loops isotopic to G corresponds to an orbit under this action. LEMMA 11. Let G be a loop with transiti¨e automorphism group and nŽ. G the number of isomorphism classes of loops isotopic to G. Ž.a If G is in class A, then n Ž G .G 5. Ž. Ž . b If G is in class B or class C s C123j C j C , then n G s 2. Ž.c If G is in class G, then nŽ. G s 1. Proof. PartŽ. c merely restates that class G consists of G-loops. ForŽ. a , note that if G is in class A, then e is fixed by each coordinate of every autotopism. Hence no two of the sets Ti lie in the same orbit under the action of ᑛŽ.G . It is easy to see that T4 is nonempty whenever the cardinality of G is at least 3. The unique loops of orders 1 and 2 are both groups and in class G, so every loop in class A has cardinality at least 3, Ž. Ž. Ž . and a follows. For b , let x123, x , x g T. Pick i such that G is not in Ž.Ž. class Ci. Then there exists ␣123, ␣ , ␣ g ᑛ G such that xii␣ s e,by Lemma 8. Hence Ž.x123, x , x lies in the same orbit as either T0or Ti. Thus there are at most two orbits. If there is only one, then G is a G-loop and must, by Corollary 4 and Lemma 6, be in class G, a contradiction. For the following classification theorem, we split class G in order to parallel the Barlotti᎐Strambach classification. THEOREM 12. Let G be a loop with transiti¨e automorphism group and let nŽ. G be the number of isomorphism classes of loops isotopic to G. G belongs LOOPS WITH TRANSITIVE AUTOMORPHISMS 223 to one of the following classes: Ž.AG is in class A if and only if nŽ. G G 5. Ž.BG is in class B if and only if nŽ. G s 2 and e¨ery left, right, or middle pseudo-automorphism is an automorphism. Ž. Ž . Ž. CG is in class C123resp., C , C if and only if n G s 2 and e¨ery element of G is a companion of a rightŽ. resp., left, middle pseudo-automor- phism. Ž. Ž. GllG will be said to be in class G if and only if n G s 1 and G is not a group. Ž.GggG will be said to be in class G if and only if G is a group which is not the additi¨e group of a ¨ector space o¨er a field. Ž. G¨¨G will be said to be in class G if and only if G is the additi¨e group of a ¨ector space o¨er a field. This theorem is essentially the Barlotti᎐Strambach classificationw 1, Corollary 17.3x , with ‘‘sharply transitive’’ replaced by ‘‘transitive.’’ The only other difference is that Barlotti and Strambach classify all loops according to whether they are isotopic to loops with sharply transitive automorphism groups, whereas we consider only the tag-loops themselves. If we restrict their classes to tag-loops, then each of our major classes isŽ. possibly larger than their corresponding classes. Using the notation ofwx 1 and denoting the class of all tag-loops by T, we may summarize the relationship as I.2 l T : A I.3 l T : B I.4 l T : C Ž.7 I.5 : Gl II.1 s Gg II.2 s G¨ . We now turn to some examples. We shall make use of the following lemma: LEMMA 13. Let G by any quasigroup and Ž.Ž.␣, , ␥ g ᑛ G . Then for all x, y g G, y1 y1 1 LxŽ. Ly Ž.s␥Lx Ž␣ .Ly Ž␣␥ .y ,8Ž. y1 y1 1 RxŽ. Ry Ž.s␥Rx Ž .Ry Ž␥ .y.9Ž. 224 ARTHUR A. DRISKO Proof. Given any z g G, write z as xa for some a. Then y1 zLŽ. x L Ž. y s ya. ŽŽ.Hence Lxy1 LyŽ.can be thought of as the map which takes each entry of the xth row of the Cayley table of G to the corresponding entry of the yth row.. Now z␥ s x␣ и a,so y1 z␥LxŽ.␣ Ly Ž.␣ sy␣иa sŽ.ya ␥ y1 sŽ.zLŽ. x L Ž. y ␥ , and hence y1 1 y1 ␥ LxŽ␣ .Ly Ž␣␥ .y sLx Ž. Ly Ž., provingŽ. 8 . The proof ofŽ. 9 is analogous. COROLLARY 14. Let G be a loop with transiti¨e automorphism group. If there exists Ž.Ž.␣ Ј, Ј, ␥ Ј g ᑛ G such that e␣ Ј / e Ž resp., such that eЈ / e ., then all maps LŽ. xy1 LŽ.Ž y resp., Rx Ž.y1 RyŽ..,with x / y are conjugates. Proof. Taking x s e and letting Ž.␣, , ␥ run through all automor- phisms of G inŽ. 8 shows that all maps Le Ž.y1 LyŽ.with y / e are 1 conjugate. If on the other hand we take x s e␣ Ј, then LeŽ.y LyŽ.is conjugate to LxŽ.y1 LyŽ.␣Ј, and as y runs through all elements of G not equal to e, y␣ Ј runs through all elements not equal to x. Finally, since x is the image of e under an autotopism, so is every element of G, by Lemma 8 and Corollary 9. The assertion for L follows, and the proof for R is similar. Ž. We can now give the promised examples. Let G5, q be the loop of order 5, with identity element 0, whose Cayley table is the following: q 01abab 001abab 110baba aaab01b bbaab01 ab ab b 1 a 0 ŽReaders familiar with neofields should note that this is the additive loop of a neofield whose multiplicative group is the Klein 4-group.. Let ␣ s Ž.Ž.1ab ab,s Ž.Ž.1 ba ab,␥s Žabab .. It is easy to check that each of these maps is an automorphism of G55,soG is a tag-loop. In fact, LOOPS WITH TRANSITIVE AUTOMORPHISMS 225 these maps generate AutŽ.G5 , so by a routine group-theoretic argument, AutŽ.G5 is the set of all even permutations ofÄ4 1, a, b, ab and is therefore isomorphic to A4. y1 We claim that G5 is in class A. Recall that the map LxŽ. LyŽ.takes the xth row of the Cayley table of G5 to the yth row. Then we easily see that y1 LŽ.0 L Ž.1 s Ž01 .Žabab ., y1 LŽ.1 La Ž.s Ž1 ab0 ab.. Since these maps have different cycle structures, they cannot be conju- gates. By Corollary 14, every autotopism of G5 fixes e in the first coordinate. A similar check shows that every autotopism of G5 fixes e in the second coordinate, so G5 is in class A. Let Ž.G6 , и be the loop of order 6 with identity e whose Cayley table is the following: и e 01234 ee01234 00e4321 112e043 2243e10 33104e2 443210e ŽThis loop, like G5, can be thought of as the ‘‘additive’’ loop of a neofield whose ‘‘multiplicative’’ group is the cyclic group of order 5.. If we let ␣sŽ.Ž.01234ands1 2 4 3 , then one can check that ␣ and  are automorphisms of G6 , so it is a tag-loop. In fact, it is not difficult to show that these two automorphisms generate AutŽ.G6 . Further- more, one can check that Ž.Ž.Ž.Ž.Ž.Ž.Ž.Ž.e 30412,01,e30412 is an autotopism of G6. Since this autotopism takes e to 3 in the first and third coordinates, G6 must, by Lemma 8 and Corollary 9, be in one of the classes B, G,orC2 . If we look at the maps y1 ReŽ. R Ž.0s Že 01423, .Ž .Ž . y1 RŽ.0 R Ž.1s Ž013 .Že24, . we see, by Corollary 14, that every autotopism of G6 fixes e in the second coordinate. Hence G62must be in class C . 226 ARTHUR A. DRISKO We can get an example of a loop in class C1 by taking the ‘‘transpose’’ of G6. It is clear what this means in terms of the Cayley table. Alge- braically, we take the quasigroup with the same elements as G6 , with a new multiplication ( given by x и y s z m y( x s z.10Ž. This quasigroup will still be a loop with the same identity, will still have tag since the automorphisms are the same, and will be in class C1. Similarly, we can get an example of a loop in class C3 by taking the ‘‘column adjugate’’ of G6 wx11 . This is defined as the quasigroup with multiplication ) given by x и y s z m x) z s y.11Ž. This will still be a loop with identity e since G6 satisfies the identity x и x s e for all x g G. Its automorphisms are the same as those of G6 , and it is in class C3. We shall have to work much harder to construct an example of a loop from class B. We shall first construct a quasigroup isotopic to such a loop and then use the properties of this quasigroup to show that the loop is in class B. Let V be a vector space of dimension 3 over F2 , the field of two Ä4 elements. Define a binary operation ) on Q7s V _ 0by xqy if x / y x) y s ½ x if x s y. Ž. This operation makes Q7, ) a quasigroup. Given a basis of V, the coordinates of an element of Q7 can be thought of as a binary number. If we label the elements of Q7 by the decimal equivalents of these binary numbersŽŽ.. e.g., 1, 0, 1 l 5 , we get the following multiplication table for Q7: ) 1234567 11325476 23216745 32137654 45674123 54761532 67452361 76543217 It is easy to see that a bijection : Q77ª Q is an automorphism if and only if the extension of to all of V gotten by taking 0 s 0isan automorphism of V, that is, an element of PSLŽ. 3, 2 . We claim that every LOOPS WITH TRANSITIVE AUTOMORPHISMS 227 Ž. autotopism of Q7 is such an automorphism. To see this, let ␣, , ␥ g Ž. Ä4 Ä 4 ᑛQ7 . Since ␣ is a bijection, for some x g 4, 5, 6, 7 the set 1␣,2␣, x␣ is a basis of V. SinceÄ4 1, 2, x is also a basis, there exists ␦ g PSLŽ. 3, 2 such that 1␣␦ s 1, 2␣␦ s 2, and x␣␦ s x. By replacing Ž.␣, , ␥ with Ž.␣␦, ␦, ␥␦ , we may assume without loss of generality that ␣ fixes 1, 2, and x. If we let ⑀ g PSLŽ. 3, 2 be the element which fixes 1 and 2 and takes xto 4, then replacing Ž.Ž␣, , ␥ with ⑀␣⑀y1,⑀⑀y1,⑀␥⑀y1., we may assume without loss of generality that ␣ fixes 1, 2, and 4. ApplyingŽ. 8 twice gives us y1 1 y1 ␥ LŽ.1 L Ž.2 ␥y s L Ž.1 L Ž.2 s Ž1325647 .Ž .Ž . y1 1 y1 ␥LŽ.1 L Ž.4␥y sL Ž.1 L Ž.4s Ž1452637. .Ž .Ž . Since ␥ commutes with both of these permutations, consideration of the cycle structures easily leads to the conclusion that ␥ is the identity permutation. Applying Ž.␣, , ␥ to the equation 1) x s y gives 1) x s y, so  is also the identity, from which it similarly follows that ␣ is the Ž.Ž. identity. Hence ␣, , ␥ g Aut Q7 , proving our claim. Now we construct the sought-after loop. Let  s Ž.Ž.Ž.234567, Ž. Ž. and define G77as the isotope of Q by , , , that is, G 7, и s Ž.Ž. Q7,),,. Then the multiplication table of G7 is и 1234567 11234567 22316745 33127654 44675123 55761432 66452371 77543216 Hence, G7 is a loop with identity 1. The autotopisms of G7 are all of the form Ž.Ž.Ž. ,  , ␣y1 ,␣,␣,,,1 Ž.3 1 where ␣ g PSLŽ. 3, 2 . Now 1␣y s1 if and only if 1␣ s 1, so anyŽ left, right, or middle. pseudo-automorphism of G7 is an automorphism, by Lemma 1. On the other hand, PSLŽ. 3, 2 is doubly transitive on the elements of Q7. This implies that the set of autotopisms fromŽ. 13 with Ž. 1␣s1, namely Aut G7 , is transitive on the elements not equal to 1, whereby G7 is a tag-loop. Furthermore, for every x g G7 there is an autotopism of G77taking 1 to x in the third coordinate. Hence G must be in class B. 228 ARTHUR A. DRISKO This example was inspired by the classification of Steiner triple systems with doubly transitive automorphism groups given by Hall inwx 8 . Q7 is equivalent to such a Steiner triple system. In fact, one can show that any loop G g B is isotopic to some quasigroup Q which is idempotent, has doubly transitive automorphism group, and has no autotopisms other than automorphisms. If Q is totally symmetricŽ i.e., satisfies the identities xy s yx and xxyŽ.sy ., then it is equivalent to a Steiner triple system with doubly transitive automorphism groupwx 15, Theorem V.1.11 . However, I do not know whether any examples of loops in class B arise from quasigroups which are not totally symmetric. Since any vector space over a field is an abelian group with transitive automorphism group, we have examples of loops in class G¨ . Neumann used free products with amalgamation to construct infinite groups G such that all nonidentity elements of G are conjugatewx 13 . For such G, then, the inner automorphisms are transitive, so we have examples of class Gg . I do not know of any examples of class Gl. Note that our examples G57g A, G g B, and G 6g C have automor- phism groups which are doubly transitive on nonidentity elements. Hence I.3 l T / A I.3 l T / B Ž.14 I.4 l T / C. Barlotti and Strambach point out that there are many examples of loops in class I.2, but they do not have examples of classes I.3, I.4, or I.5, so we are left with several questionsŽ cf.wx 1, pp. 99᎐102. . Are the classes I.3 and I.4 empty? In other words, do there exist loops in classes B and C with sharply transitive automorphism groups? Is Gl empty, or are there tag-loops which are G-loops but not groups? If Gl is nonempty, does it contain Ž.only loops with sharply transitive automorphism groups? ACKNOWLEDGMENT The author thanks Professor George Bergman at the University of California at Berkeley for many helpful comments on earlier versions of this article. REFERENCES 1. A. Barlotti and K. Strambach, The geometry of binary systems, Ad¨. Math. 49 Ž.1983 , 1᎐105. 2. V. D. Belousov, ‘‘On the Structure of Distributive Quasigroups,’’ Izdat. Nauka, Moscow, 1967.wx in Russian LOOPS WITH TRANSITIVE AUTOMORPHISMS 229 3. R. H. Bruck, Some results in the theory of quasigroups. Trans. Amer. Math. Soc. 55 Ž.1944 , 19᎐52. 4. R. H. Bruck, Loops with transitive automorphism groups, Pacific J. Math. 1 Ž.1951 , 481᎐483. 5. R. H. Bruck, ‘‘A Survey of Binary Systems,’’ Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 20, Springer-Verlag, BerlinrNew York, 1971. n 6. A. A. Drisko, Loops of order p q 1 with transitive automorphism groups, Ad¨. Math., to appear. 7. T. Evans, On multiplicative systems defined by generators and relations. II. Monogenic loops, Proc. Cambridge Philos. Soc. 49 Ž.1953 , 579᎐589. 8. M. Hall, Jr., Steiner triple systems with doubly transitive automorphism groups, J.Combin. Theory Ser. A 38 Ž.1985 , 192᎐202. 9. D. F. Hsu and A. D. Keedwell, Generalized complete mappings, neofields, sequenceable groups and block design, I, Pacific J. Math. 3 Ž.1984 , 317᎐332. 10. D. R. Hughes, Planar division neo-rings, Trans. Amer. Math. Soc. 80 Ž.1955 , 502᎐527. 11. J. C. M. Janssen, On even and odd latin squares, J. Combin. Theory Ser. A 69 Ž.1995 , 173᎐181. 12. A. D. Keedwell, The existence of pathological left neofields, Ars Combin. 16B Ž.1983 , 161᎐170. 13. B. H. Neumann, An essay on free products of groups with amalgamations, Philos. Trans. Roy. Soc. London Ser. A 246 Ž.1954 , 503᎐554. 14. L. J. Paige, Neofields, Duke Math. J. 16 Ž.1949 , 39᎐60. 15. H. O. Pflugfelder, ‘‘Quasigroups and Loops: Introduction,’’ Sigma Series in Pure Mathe- matics, Vol. 7, Heldermann Verlag, Berlin, 1990.