U.U.D.M. Project Report 2016:18

Examensarbete i matematik, 15 hpHandledare: Anders Öberg Examinator: Veronica Crispin QuinonezJuni 2016

Department of MathematicsUppsala University

Moufang LoopsGeneral theory and visualization of non-associative Moufang loops of order 16

Mikael Stener

Moufang LoopsGeneral theory and visualization of

non-associative Moufang loops of order 16

Thesis by: Mikael Stener

Supervisor: Anders ObergUppsala University

Department of Mathematics

April 1, 2016

Abstract

This thesis examines the algebraic structure of non-associative Moufang loops.We describe their basic properties such as their alternativity and flexibility. Aproof of Moufang’s Theorem is presented which implies the important notice ofdi-associativity. We then provide a case study which leads us to find all non-associative Moufang loops of order less than 32. We study in particular the onesof order 16 and provide a visualization of the multiplication in these loops ashas been previously done by Vojtechovsky [6] with the (only) non-associativeMoufang loop of order 12.

A brief presentation of the life and work of the name giver of Moufang loops,Ruth Moufang, is also given.

Contents

1 Introduction 2

2 Theory 42.1 Groupoids, quasigroups and loops . . . . . . . . . . . . . . . . . . 42.2 Inverse property and autotopisms . . . . . . . . . . . . . . . . . . 82.3 Moufang loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Moufang’s Theorem 13

4 Moufang loops of small order 18

5 Visualization of Moufang loops of order 16 275.1 M16(D4, 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.2 M16(Q, 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.3 M16(C2 × C4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.4 M16(C2 × C4, Q) . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.5 M16(Q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

6 Appendix 356.1 Case studies of elements of different order in Moufang loops . . . 356.2 Cayley tables for Moufang loops of order 16 . . . . . . . . . . . . 42

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Chapter 1

Introduction

Since this thesis arose from an interest in wanting to achieve a general under-standing, or ”feeling” of the concept of Moufang loops, the purpose is also togive the reader this same feeling. A complete answer to the question of whatMoufang loops are will be given in the Theory section of this thesis, but a briefanswer is that Moufang loops are groups that are not necessarily associative butmust satisfy the Moufang identity (xy)(zx) = x((yz)x) for all x, y, z in the loop.For the reader familiar with group theory, Figure 1.1 will help put Moufangloops in a context with other known algebraic structures1. We notice firstlythat the higher up in the chain, the more general the structure. Also, all groupsare Moufang loops, which are all loops, which are all quasigroups, which are allgroupoids, but the opposite statements are not true. In this thesis, it is the leftside of Figure 1.1 that is of interest.

Figure 1.1: Algebraic structures between groupoids and groups

Monoids

Semigroups

Groupoids

Groups

Moufang Loops

Loops

Quasigroups

invertibility

identity

associativity

associativity

Moufang identity

identity

divisibility

The name of Moufang loops stems from the german mathematician Ruth

1This image is inspired by the Wikipedia article on Quasigroups

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Moufang, born in Darmstadt in 1905. Having developed an interest for math-ematics in high school, she later continued on this path at the University ofFrankfurt where she got her Ph.D. in mathematics at an age of 25. Receiving alectureship in both Knigsberg and Frankfurt, the logical course of events wouldhave been to become a Privatdozent, a lecturer recognized by the university butreceiving no formal salary for it. However, the rise of the Third Reich made itimpossible for her to receive this title due to the state’s view on female lead-ership. After spending the war years at a industry research institute, she wasgiven a position as Privatdozent in 1946, and a full professorship in 1957, bothat the University of Frankfurt. She died in Frankfurt in 1977 [3].

Moufang’s early work was mostly on projective planes and her later yearswas dedicated to theoretical physics. The focus of this thesis is however on theresults of the paper Moufang wrote in 1935: Zur Struktur von Alternativkorper,translated to ”On the structure of alternative division rings”.

In this paper, Moufang studies the non-zero elements of an alternative divi-sion ring. These elements form a loop which were later named Moufang loopsby Bruck [2]. In Zur Struktur von Alternativkorper, Moufang proves what iscalled Moufang’s Theorem stating that any three elements a, b, c in a Moufangloop that associate, i.e. where a(bc) = (ab)c, generate a group.

*

This thesis first provides the theory for groupoids in general and loops inparticular. The definitions and theorems provided here are both for the use ofunderstanding the subsequent chapters but also for providing the reader with abroader sense of the concept of Moufang loops. Chapter 3 is then dedicated tothe important Moufang’s Theorem. We then continue with finding the small-est non-associative Moufang loops by case studies in Chapter 4 and eventuallypresent visualizations of the multiplication in non-associative Moufang loops oforder 16 in Chapter 5.

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Chapter 2

Theory

The reader of this thesis is assumed to have knowledge of the basic conceptsof groups, so comparisons between groups and other structures will be maderepeatedly.

Regarding notation, a binary operation might be noted with the symbolfirst defined (usually (·)), but will often instead be written as juxtaposition, forsimplicity. Juxtaposition have higher priority than (·).

Almost all theory in this chapter is gathered from the textbook Quasigroupsand Loops - Introduction by Pflugfelder [1], unless stated otherwise.

2.1 Groupoids, quasigroups and loops

All groups or group-like structures do not only constitute of elements. It isnot the elements themselves that are of most interest, but rather how theseelements relate with respect to the binary operation of the structure. A binaryoperation on a non-empty set G is formally defined as a map α : G × G → G.So any set equipped with a binary operation is by definition closed under thisoperation. We shall now meet the first structure constituting of a set and abinary operation, namely groupoids, the most general structure shown in Figure1.1.

Definition 2.1. A groupoid (G, ·) (sometimes also called magma) is a non-empty set G equipped with a binary operation (·). The order of the groupoid issimply the cardinality |G| of the set G.

A groupoid (G, ·) may be noted simply by G, where the binary operation isassumed to be noted by (·) or by juxtaposition.

To get a sense of just how general groupoids are, we show in Table 2.1 theCayley tables (often called multiplication table) of the groupoids of order 2where all are different up to isomorphism. We notice in particular that onlyone of the tables is a latin square, i.e. that all elements occur only once in eachrow and column. In the first groupoid for instance, both 1 · 1 = 1 and 1 · 2 = 1.

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Table 2.1: Cayley tables of groupoids of order 2

· 1 21 1 12 1 1

· 1 21 1 12 1 2

· 1 21 1 12 2 1

· 1 21 1 12 2 2

· 1 21 1 22 1 1

· 1 21 1 22 1 2

· 1 21 1 22 2 1

· 1 21 1 22 2 2

This is of course impossible for groups, and we will see soon that this propertydisappears quickly as we move downwards on the left side of Figure 1.1.

An important concept when studying non-associative structures like loopsare the translation maps La and Ra. These are defined as

La : G→ G, La(x) = a · xand

Ra : G→ G, Ra(x) = x · a.

Why not simply write left or right multiplication by element a? To understandthe reason for this and for future importance, we will study these maps inmore detail. Firstly, it is common that La(x) is written as an element in itself,La ∈ Sn, where Sn is the symmetric group, since the map can be seen asa permutation. The composition of two translation maps LaRb acting on anelement x in a groupoid is read from right to left and thus equivalent to a(xb)since the composition implies that we first multiply with the element b on theright and then multiply with the element a on the left. Similarly, for example,we have that RbLbLaRa acting on x is equivalent to (b(a(xa)))b. It shouldnow be evident that translation maps are of importance when dealing withnon-associative structures.

We now continue to the next structure on the left side of Figure 1.1, quasi-groups.

Definition 2.2. A quasigroup is a groupoid G where the maps La : G→ G andRa : G→ G are bijections for all a ∈ G.

Another way of looking at this definition is that given two of x, y, z as ele-ments in a quasigroup G, the third can be selected uniquely so that x · y = z.Looking back at Table 2.1, we see that most of these groupoids do not havethis property. In fact, the only one is the one that is also a group, namelyC2. It should also be clear that quasigroups satisfy the cancellation laws; botha · x = a · y and x · a = y · a imply x = y.

Definition 2.2 implies that there is an inverse map, noted L−1a such thatLa(x) = y implies L−1a (y) = x and that the equivalent holds for Ra. Theseinverse maps are called the conjugates of quasigroups. We use these to definetwo new binary operations as

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x\y := L−1x (y) and x/y := R−1y (x).

Due to the bijectivity of the translation maps in quasigroups, the conjugatesare unique. It is clear that (G, \) and (G, /) are quasigroups.

To connect to the terminology of Figure 1.1, quasigroups can be definedas groupoids where every element is uniquely divisable with another, wheredivisable in this sense means that a ·x = y implies x = a\y and x ·a = y impliesx = y/a. The similarity to ordinary multiplication and division is clear.

We can make use of the translation maps when defining other known con-cepts. For instance,

Definition 2.3. A groupoid G is called commutative if La = Ra for all a ∈ G.

Definition 2.4. A groupoid G is called associative if Ra·b = RaRb for alla, b ∈ G.

Definition 2.3 implies that ax = xa for all x and a in G which is the commonway to define commutativity, and Definition 2.4 implies that ab · x = a · bx forall a, b, x ∈ G.

We now come to the notion of identity elements, which we also define withthe help of translation maps.

Definition 2.5. Let G be a groupoid and let e ∈ G. Then e is called a leftidentity element for G if Le is the identity map of G, meaning that Le(x) = xfor all x ∈ G. If e is such that Re is the identity map of G, then e is called aright identity element. If e is both a left and right identity element, then we saythat e is an identity element.

It will not be proved here, but it should be obvious that if a groupoid hasboth a left and a right identity element, then these are the same element.

We are now ready to define the next structure in Figure 1.1, namely loops.

Definition 2.6. A loop is a quasigroup with an identity element.

As an example, we take a look at a non-associative loop of order 5. LetG = {1, 2, 3, 4, 5} and let (·) be given by the following Cayley table.

· 1 2 3 4 51 1 2 3 4 52 2 1 4 5 33 3 5 1 2 44 4 3 5 1 25 5 4 2 3 1

The definition of quasigroups implies that the Cayley table of a quasigroup isa latin square and we see that the elements of G occur exactly once in each rowand column so (G, ·) is indeed a quasigroup. We also see that 1 is the identity

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element of G. However, since 3 · (3 · 4) = 3 · 2 = 5 6= 4 = 1 · 4 = (3 · 3) · 4, G isnot associative, and thus not a group.

Speaking of groups, we are now at the bottom of Figure 1.1 and we heredefine groups as

Definition 2.7. A group is a loop that is associative.

In Figure 1.1, we have now moved from quasigroups via the existence ofidentity elements to loops and from loops via associativity to groups. We willlater come to the core of this thesis, Moufang loops, but we first ask ourselveswhether we can go from quasigroups via associativity to some other structure,i.e. are there associative quasigroups that have no identity element and aretherefore neither loops nor groups? The following theorem answers this question.

Theorem 2.1. If G is a quasigroup which is associative, then G necessarily hasan identity element and is thus a group.

Proof. We have from the definition of groupoids that G is non-empty, so thereis an element a ∈ G. From the definition of quasigroups and the commentsfollowing it, there is an element e ∈ G such that a · e = a. Let now b be anyelement in G. Again, we have that there is an element y ∈ G such that y ·a = b.Due to the cancellation laws of quasigroups and the assumed associativity, wehave Re(b) = be = ya · e = y · ae = ya = b. Since b is any element in G, we thushave that Re is the identity map on G.

Again, let b be any element in G. Then bb = be ·b = b ·eb due to associativity.By left cancellation, we have that bb = b · eb implies that b = eb = Le(b) and soLe is also the identity map on G and by definition, e is an identity element ofG. This element is unique since if e′ is also an identity element, then ee′ = eand ee′ = e′ and so e = e′. G is hence a group.

Definition 2.8. A non-empty subset H of a set G is called a subquasigroup/subloop/subgroup of a quasigroup (G, ·) when (H, ·) is a quasigroup/loop/group.

When later studying Moufang loops, we will notice that many of the subloopsof the loops are in fact groups and can thus be called subgroups.

A very important concept in group theory is the center of the group. Thisconcept is present in groupoids in general and we will define this soon, but wefirst need another concept very important for non-associative structures, namelythe nucleus.

Definition 2.9. The left nucleus Nλ, the middle nucleus Nµ and the rightnucles Nρ of a groupoid G are defined as

Nλ(G) = {a ∈ G | a · (x · y) = (a · x) · y for all x, y ∈ G}Nµ(G) = {a ∈ G | (x · a) · y = x · (a · y) for all x, y ∈ G}Nρ(G) = {a ∈ G | (x · y) · a = x · (y · a) for all x, y ∈ G}

The nucleus of G is defined as

N(G) = Nλ(G) ∩Nµ(G) ∩Nρ(G)

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Similar to how the center in non-abelian groups contain the elements thatdo commute with all other elements, the nucleus of non-associative groupoidscontain the elements that do associate with all other pair of elements. We cannow define the center of groupoids.

Definition 2.10. Let G be a groupoid. The center of G is given by

Z(G) = {a ∈ N(G) |La = Ra}

So in groupoids in general, an element is in the center both if it commutesand associate with all other elements.

We note that the identity element of a group will always be in the center, buta groupoid might not have an identity element, and so the center of a groupoidmight be empty. We also see that if a groupoid G is associative, then N(G) = Gand if it is also commutative, then Z(G) = G.

We can now ask ourselves to which extent elements that are not in the centercommute with each other. If neither a nor b are in Z of a groupoid G, thenab 6= ba, but ab = ba · c for some element c. If G is a quasigroup, then c isunique. We call this element the commutator of a and b and we denote it [a, b].

Similarly, in non-associative quasigroups where a, b, c are not in the nucleus,we have a · bc 6= ab · c but we have a · bc = (ab · c) · d for some unique elementd. This element is called the associator of the quasigroup and we denote it by[a, b, c] (the order is important, we do not have [a, b, c] = [a, c, b] in general)[2].

2.2 Inverse property and autotopisms

A quasigroup G is said to have the left inverse property and called a L.I.Pquasigroup (or L.I.P. loop if it is a loop) if there exists a bijection δλ : x 7→ xλ

such that xλ · xy = y for all x and y in G. A quasigroup/loop is said to bea R.I.P quasigroup/loop (a quasigroup/loop with the right inverse property) ifthere exists a bijection δρ : x 7→ xρ such that yx · xρ = y for all x and y in G.

A quasigroup/loop that has both the L.I.P. and the R.I.P. is said to havethe inverse property and is called an I.P. quasigroup/loop. An I.P. loop thussatisfies x−1 ·xy = y = yx ·x−1 where x−1 is, as in groups, called the inverse ofx. This property is very important and will play a dominant role when provingMoufang’s Theorem in chapter 3.

One of the axioms of groups is that all elements have inverses, so that forall a ∈ G where G is a group there exists an element denoted a−1 such thata · a−1 = e where e is the identity element of the group. This is of course nottrue for groupoids in general, but for any loop L, we at least have that there forall x ∈ L exists a unique element denoted xλ that solves the equation xλx = e.This is then called the left inverse of x. Similarly, the right inverse xρ of anelement x solves the equation xxρ = e.

Any quasigroup can in general have either the right inverse property or theleft inverse property. However, the next theorem shows that if a loop has oneof them, it has the other.

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Theorem 2.2. If a loop L is an L.I.P. loop or an R.I.P. loop then aλ = aρ =a−1.

Proof. Let 1 be the identity element of an L.I.P. loop L. Then aaρ = 1, aλa = 1,aλ(aaρ) = aρ and aλ(aaρ) = aλ · 1 = aλ so aλ = aρ = 1. Similarly, if L hasR.I.P. then aρ = (aλa)aρ = aλ.

A concept that will later be used frequently in the proof of Moufang’s theo-rem is autotopisms.

Definition 2.11. An autotopism of a quasigroup G is a triple (α, β, γ) of per-mutations of G where α(x)β(y) = γ(xy) for all x, y ∈ G.

Autotopisms can, as well as translation maps, be used to express identityrelations of quasigroups. For example, we have that the associative law can beexpressed as (ι, Rz, Rz), where ι is the identity map, since ι(x)Rz(y) = x(yz) =(xy)z = Rz(xy).

We now prove a theorem that later will be used in proving properties ofMoufang loops.

Theorem 2.3. Let σ = (α, β, γ) be an autotopism of an I.P. loop L and let δbe defined as δ(x) = x−1. Then σλ = (δαδ, γ, β), σµ = (γ, δβδ, α) and σρ =(β, α, δγδ) are also autotopisms of L.

Proof. The autotopism (α, β, γ) implies the identical relation α(x)β(x) = γ(xy).Since L is an I.P. loop and thus a R.I.P. loop, this relation can be written as

α(x) = γ(xy) · δβ(y) (2.1)

We put xy = a, y = b−1 = δ(b) and thus x = ab. Substituting a and b in(2.1) we get γ(a) · δβδ(b) = α(ab) for all a, b ∈ L. Thus σµ = (γ, δβδ, α) is anautotopism on L. In the same way, the other statements of the theorem can beproved.

2.3 Moufang loops

We come now finally to the central concept of this thesis, Moufang loops. WhenRuth Moufang wrote Zur Struktur von Alternativkorper she called them Quasi-groups and defined a Quasigroup Q* as a set with multiplication such that

(I) for any two elements x, y there exists a unique product xy

(II) there exists an identity element 1, and to any element x, there exists aunique element x−1 such that x−1x = 1 = xx−1

(III) for any x and y we have x · x−1y = xx−1 · y and yx−1 · x = y · x−1x

(IV) for any x, y, z we have (x(zx))y = x(z(xy))

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Using the terminology of today, we see that (III) in fact means that Q* hasthe inverse property. Although Moufang used this property as a definition, wewill later use another definition for Moufang loops and then prove that it hasthe inverse property.

Moufang then defined a Quasigroup Q** as a set that satisfies the aboveconditions, and also satisfies (xy)(zx) = x((yz)x). In the same paper, sheshowed that (IV) is equivalent to both ((xz)x)y = x(z(xy)) and to ((yx)z)x =y(x(zx)). It was later proved that

(x(zx))y = x(z(xy)) (M1)

(xy)(zx) = x((yz)x) (M2)

((xz)x)y = x(z(xy)) (M3)

((yx)z)x = y(x(zx)) (M4)

in fact all are equivalent and that Q* and Q** thus represent the same structure.These structures are today called Moufang loops and we will here define it withanother identity.

Definition 2.12. A loop (M, ·) is called a Moufang loop if it satisfies the fol-lowing identity:

(xy)(zx) = (x(yz))x (M5)

We will soon prove the equivalence of the so called Moufang identities (M1)-(M5), but we first define other important concepts of loops.

Definition 2.13. A loop is said to be left alternative, flexible and right alter-native, respectively if it satisfies the following identities, respectively:

x · xy = xx · yxy · x = x · yxyx · x = y · xx

If a loop is both left and right alternative, then it is called alternative

It is evident that alternativity and flexibility is a weaker form of associativitysince a · bc = ab · c is true if any two of a, b and c are equal.

The observant reader noticed that Ruth Moufang’s article on Moufang loopswas translated to ”On the structure of alternative division rings” and that Mo-ufang loops therefore should be alternative. This is true and we will prove thisas well as the equivalence of (M1)-(M5) and the inverse property of Moufangloops in the same theorem.

Theorem 2.4. Each Moufang loop M is left and right alternative, flexible andsatisfies the inverse property. Also, the Moufang identities are equivalent.

Proof. The proof is divided into nine steps (i)-(ix).

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(i) yλ · yx = x (L.I.P.)

(ii) xy · x = x · yx (flexible)

(iii) (M2) and (M5) are equivalent

(iv) xy · y−1 = x (R.I.P.)

(v) (M4) and (M5) are equivalent

(vi) (M3) and (M4) are equivalent

(vii) xx · y = x · xy (left alternative)

(viii) xy · y = x · yy (right alternative)

(ix) (M1) and (M5) are equivalent

(i): From (M1) we have yλy · zyλ = (yλ · yz)yλ ⇐⇒ zyλ = (yλ · yz)yλ anddue to cancellation we have yλ · yz = z which is the left inverse property.

(ii): By setting y = 1 in (M5) we get x · zx = xz · x.

(iii): From (ii), we see that the right sides of (M2) and (M5) are equivalent,so that (M2) and (M5) are equivalent.

(iv): From (M2) we have xy = xy ·x−1x = x(yx−1 ·x) and thus y = (yx−1)x.From (i) and (iv) we now have that each Moufang loop has the inverse property.

(v): We can write (M5) as an autotopism σ = (Lx, Rx, LxRx). Then, by The-orem 2.3, σλ = (δLxδ, LxRx, Rx), where δ(x) = x−1, is also an autotopism of aMoufang loop M . Thus δLxδ(yx)·LxRx(z) = Rx((yx)z) which can be written as(x(yx)−1)−1((xz)x) = ((yx)z)x. The left side is now (x(x−1y−1))−1((xz)x) =(y−1)−1((xz)x) = y((xz)x) and so y((xz)x) = ((yx)z)x. Now applying (iii) tothe left side, we get y(x(zx)) = ((yx)z)x which is (M4).

(vi): Taking inverses on both sides of (M4) we get

(((yx)z)x)−1 = (y(x(zx)))−1 ⇐⇒ x−1((yx)z)−1 = (x(zx))−1y−1 ⇐⇒x−1(z−1(yx)−1) = ((zx)−1x−1)y−1 ⇐⇒ x−1(z−1(x−1y−1)) = ((x−1z−1)x−1)y−1

and now replacing x−1, y−1, z−1 with x, y, z, respectively, we get x(z(xy)) =((xz)x)y which is exactly (M3).

(vii): Putting z = 1 in (M3) we get y · yx = yy · x, the left alternative law.

(viii): From (vii) we get (y(yx))−1 = ((yy)x)−1 ⇐⇒ (x−1y−1)y−1 =x−1(y−1y−1) and substituting x−1, y−1 with x, y respectively, we have the rightalternative law.

(ix): Applying the flexible law to the left side of (M1), we get ((xz)x)y =x(z(xy)) which is (M3) and due to (vi) and (v), (M1) is thus equivalent to(M5).

For future importance, we also need to mention that for Moufang loops M ,there might be elements that commute with all other elements in the loop,but who do not associate with all other elements. In this case, these elements

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are said to be in the Moufang center of M , denoted C(M). In fact there arecommutative non-associative Moufang loops, the smallest of which is of order81, and in this case C(M) = M but Z(M) 6= M [1].

We have now seen some of the properties that Moufang loops possess, buthave not seen any examples of non-associative Moufang loops. Of course, asstated in the introduction, all groups are Moufang loops, which should be ob-vious considering the definition, but these are all associative. What about thefollowing loop of order 5 we have seen before? Is it Moufang?

· 1 2 3 4 51 1 2 3 4 52 2 1 4 5 33 3 5 1 2 44 4 3 5 1 25 5 4 2 3 1

An easy way to see this is in the same way as we saw that it is not a group:3 · (3 ·4) = 3 ·2 = 5 6= 4 = 1 ·4 = (3 ·3) ·4, so the loop is not alternative, and canthus not be a Moufang loop. As we shall see, the smallest Moufang loop is infact of order 12, and more studies of non-associative Moufang loops will come.

One of the most important property of Moufang loops was proven by RuthMoufang in her mentioned paper and is here proved in the next chapter.

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Chapter 3

Moufang’s Theorem

First a reminder on notation; for any set of elements a1, . . . , an in a loop L wewrite 〈a1, . . . , an〉 as the subloop of L that is generated by a1, . . . an, meaning thesubloop that can be constructed only by multiplication and inverses of a1, . . . an.

In Zur Struktur von Alternativkorper, Ruth Moufang proved that if the ele-ments a, b, c in a Moufang loop M associate, i.e. that a · bc = ab · c, then 〈a, b, c〉is a group. In a way, this is intuitive; if elements associate in an otherwise non-associative structure, then all combinations of multiplication of these elementsshould associate and they should therefore generate a group. Intuitive or not,the proof is a bit tedious and the original proof is a bit technical. Here, theproof follows the outline of Ales Drapal [4] which uses nothing else than theconcepts already shown in the last chapter.

We note first that any I.P. loop satisfies (yx)−1 = x−1y−1 due to

y = yx · x−1 ⇐⇒ (yx)−1y = x−1 ⇐⇒ (yx)−1 = x−1y−1

and also that R−1a = Ra−1 and L−1a = La−1 , due to

Ra(x) = xa = y ⇐⇒

{x = ya−1 = Ra−1(y)

x = y/a = R−1a (y)=⇒ Ra−1 = R−1a

La(x) = ax = y ⇐⇒

{x = a−1y = La−1(y)

x = a\y = L−1a (y)=⇒ La−1 = L−1a

In the lemmas below we will assume that α(1) = 1 where α is some permu-tation of a loop M . For an autotopism (α, β, γ) we see that this assumptionleads to α(1)β(x) = γ(1 · x) and thus β(x) = γ(x). We also always have thatγ(x) = α(x) · b where b = β(1). Thus, whenever α(1) = 1, we can find anautotopism (α, β, γ) by putting β = γ and we then have

β(x) = γ(x) = α(x) · b (3.1)

Autotopisms form a group under composition where

(α, β, γ)(α′, β′, γ′) = (αα′, ββ′, γγ′)

13

Lemma 3.1. Let (α, β, γ) be an autotopism of an I.P. loop L. Suppose thatα(1) = 1 and that α(x) = x for some x ∈ L. Then α(x−1) = x−1 as well.

Proof. For any autotopism of an I.P. loop we have α(x)β(x−1) = γ(1). Usingα(x) = x and (3.1) we get x · α(x−1)b = b where b = β(1). So α(x−1)b = x−1band α(x−1) = x−1.

Lemma 3.2. Let M be a Moufang loop with an autotopism (α, β, γ) such thatα(1) = 1. Then if x, y ∈ M are such that α(x) = x and α(y) = y, thenα(xyx) = xyx.

Proof. Again, put b = β(1). From (3.1) we have that α(xy · x) · b = γ(xy ·x) = α(xy)β(x) = α(xy) · α(x)b = α(xy) · xb and thus α(xy · x) = (α(xy) ·xb)b−1. We have that α(xy)b = γ(xy) = α(x) · α(y)b = x · yb, so (α(xy) ·xb)b−1 = (((x · yb)b−1)xb)b−1. Now, using that M is Moufang, we get that(((x · yb)b−1)xb)b−1 = (x · yb)(b−1 · xb · b−1) = (x · yb)(b−1x) = x(yb · b−1)x =xyx.

We put X as a generating set of a Moufang loop M . Since every Moufangloop is an I.P. loop, every element in M can be expressed as multiplication ofthe elements in X± = {x, x−1;x ∈ X}. Let us now define an element in M asl(x1, . . . , xk) which is equal to 1 if k = 0 and equal to x1l(x2, . . . , xk) if k ≥ 1. Forexample, l(x1, x2, x3, x4) = x1 · l(x2, x3, x4) = x1(x2 · l(x3, x4)) = x1(x2(x3x4)).Similarly, r(x1, . . . , xk) is equal to 1 if k = 0 and equal to r(x1, . . . , xk−1)xk ifk ≥ 1, so r(x1, x2, x3, x4) = ((x1x2)x3)x4.

Lemma 3.3. Let M be a Moufang loop generated by a set X such that l(x1, . . . , xk) =r(x1, . . . , xk) for all finite sequences x1, . . . , xk over X±. Then M is a group.

Proof. Since the difference between a Moufang loop and a group is that a Mo-ufang loop is not necessarily associative, we need to show that given l(x1, . . . , xk) =r(x1, . . . , xk), a · bc = ab · c for all elements a, b, c ∈ M . To arrive at this, weshall first prove that

l(u1, . . . , un) · l(v1, . . . , vm) = l(u1, . . . , un, v1, . . . , vm) (3.2)

for all ui, vj ∈ X±. We do this by induction on n.As base case, for n = 0 and n = 1, (3.2) becomes l(v1, . . . , vm) = l(v1, . . . , vm)

and u1l(v1, . . . , vm) = l(u1, v1, . . . , vm), respectively, which are both obviouslytrue. For n ≥ 2, we put x = u1, s = l(u2, . . . , un) and t = l(v1, . . . , vm). Theinduction assumption now is that s · t = l(u2, . . . , un, v1, . . . , vm) and we willprove that if this is true, then xs · t = l(u1, . . . , un, v1, . . . , vm). We express xs · tas xs · (tx−1 · x) = x(s · tx−1)x due to Moufang identity (M2) and flexibility.Now, tx−1 = r(v1, . . . , vm)x−1 = r(v1, . . . , vm, x

−1) = l(v1, . . . , vm, x−1) and so

xs · t = x(s · l(v1, . . . , vm, x−1))x. And so, by the induction assumption, we have

xs · t = xl(u2, . . . , un, v1, . . . , vm, x−1)x

= x(r(u2, . . . , un, v1, . . . , vm)x−1)x = xl(u2, . . . , un, v1, . . . , vm)

= l(u1, u2, . . . , un, v1, . . . , vm).

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We can now show that ab·c = a·bc by setting a = l(u1, . . . , un), b = l(v1, . . . , vm)and c = l(w1, . . . , wp). The result follows since the proved equality above yields

ab · c = l(u1, . . . , un, v1, . . . , vm, w1, . . . , wp) = a · bc

Thus S = {l(u1, . . . , un);u1, . . . , un ∈ X±} is a subsemigroup of M , generatedby X±. S is also a group since all generating elements have inverses. ThusS = M .

Lemma 3.4. Let x and y be elements of a Moufang loop M. Then L−1xy LxLy =[R−1x , Ly] and R−1xyRxRy = [L−1x , Ry].

Proof. The first identity states that R−1x Ly = LyR−1x ·L−1xy LxLy. Cancelling Ly

and rewriting gives LxRxLy = LxyRx which, acting on an element z ∈M givesx((yz)x) = (xy)(zx), which is Moufang identity (M2), so the first equality is truein any Moufang loop. As for the other identity to be proved, we have L−1x Ry =RyL

−1x · R−1xyRxRy ⇐⇒ RyxLx = RxLxRy ⇐⇒ (xz)(yx) = (x(zy))x and we

have Moufang identity (M5).

For the sake of next lemma, we notice that if α = [R−1x , Ly] we have α(1) =1 because [R−1x , Ly](1) = L−1xy LxLy(1) = L(xy)−1LxLy(1) = (xy)−1(xy) = 1.Therefore, there exist β and γ such that (α, β, γ) is an autotopism.

Lemma 3.5. Let M be a Moufang loop. Suppose that α = [Rx, Ly]±1 wherex, y ∈ M or that α = Lε1x1

. . . Lεnxnwhere x1, . . . , xn ∈ M and ε1, . . . , εn ∈ −1, 1.

Then there exist β and γ such that (α, β, γ) is an autotopism of M .

Proof. We have that Rx = R−1x−1 and Lx = L−1x−1 so together with Lemma 3.4we have an autotopism if α = [Rx, Ly]±1. To prove the statement for the caseα = Lε1x1

. . . Lεnxnwe use that the Moufang identity xy · zx = x(yz ·x) states that

(Lx, Rx.LxRx) is an autotopism for every x ∈ M . So we have an autotopismwhen α = Lx and then of course when α = L−1x . An autotopism for whenα = Lε1x1

. . . Lεnxnfor arbitrary n can thus be obtained as the composition of

autotopisms (Lxi, Rxi

, LxiRxi

)εi where 1 ≤ i ≤ n.

Lemma 3.6. Let x1, x2, x3 be elements of a Moufang loop M . If x1 · x2x3 =x1x2 · x3, then

xε1σ(1) · xε2σ(2)x

ε3σ(3) = xε1σ(1)x

ε2σ(2) · x

ε3σ(3)

for all permutations σ ∈ S3 and all εi ∈ {−1, 1}, i ∈ {1, 2, 3}. Furthermore,x1 · x2x3 is equal to x1x2 · x3 whenever xi = xj, 1 ≤ i < j ≤ 3.

Proof. The latter claim simply states that each Moufang loop is flexible andleft and right alternative which already has been proven. Let us now assumethat M is generated by x = x1, y = x2 and z = x3 and that x · yz = xy · z.By Lemmas 3.4 and 3.5, any of the mappings L−1xy LxLy, R−1yz RzRy and [Rx, Lz]can be put as the mapping α in Lemma 3.1. Therefore x · yz−1 = xy · z−1,x−1 · yz = x−1y · z and x · y−1z = xy−1 · z respectively. We have thus shownthat, given any permutation σ ∈ S3 and some ε1, ε2 and ε3, the equality in the

15

lemma hold for the same σ and for any other ε′i ∈ {−1, 1}. We now need toshow that it also holds for all other permutations σ′ ∈ S3 given σ. We do thisby showing that it holds for any two permutations that generate S3, namelya 3-cycle and a transposition. The identity L−1xy LxLy = [R−1x , Ly] in Lemma3.4 means that x · yz = xy · z implies y · zx−1 = yz · x−1 and due to the firstpart of this lemma, this implies y · zx = yz · x, so we have a 3-cycle. For ourtransposition, we use the fact that each Moufang loop has the inverse propertyand thus (x · yz)−1 = (xy · z)−1 ⇐⇒ z−1y−1 ·x−1 = z−1 · y−1x−1 which, againby the first part of this lemma, implies zy · x = z · yx. So the equality that isto be proven holds for permutations σ = (1 2 3) and τ = (1 3) who togethergenerate S3.

Proposition 3.1. Let M be a Moufang loop generated by X = {x, y, z}. If x ·yz = xy·z, then u0·l(u1, . . . , uk)uk+1 = u0l(u1, . . . , uk)·uk+1 and l(u1, . . . , uk+1) =r(u0, . . . , uk+1) for any sequence u0, . . . , uk+1 of elements of X±, k ≥ 1.

Proof. There are two equalities to be proved, which both will be proved by in-duction on k. The case k = 1 follows from the assumption, since both equalitiescontain three elements. For the induction step, we first consider the equality

u0 · l(u1, . . . , uk)uk+1 = u0l(u1, . . . , uk) · uk+1.

If u0 = uk+1 then the equality holds due to the second statement of Lemma3.6 and the induction axiom. We can thus assume that, say, u0 = x anduk+1 = y. We will now prove the equality for the different cases of u1 and uk,namely u1, uk ∈ {x, y, z}±1.

We put s = l(u2, . . . , uk). The equality is then equivalent to x(u1s · y) =(x ·u1s)y. If u1 = x−1, then x(x−1s ·y) = (x ·x−1s)y ⇐⇒ x(x−1s ·y) = sy ⇐⇒x−1s · y = x−1 · sy and the last equality holds by the induction assumption. Ifinstead u0 = x−1 and u1 = x, we have x−1(xs · y) = (x−1 · xs)y, and dueto Lemma 3.6 we have x(xs · y) = (x · xs)y, viewing xs as one element. Wehave shown that the equality holds for u1 = x±1. By the induction assumption,l(u1, . . . , uk) = r(u1, . . . , uk) so the situation is left-right symmetric. The caseuk = y±1 can thus be shown by mirror reasoning.

By the induction assumption, xs · y = x · sy, and by Lemma 3.6, this impliesxy · s = x · ys. Right-multiplying by y, we get (x · ys)y = (xy · s)y = x(ys · y) bya Moufang identity. Again by Lemma 3.6 we have x · sy)y−1 = x(ys · y−1, andso we have also (x · y−1s)y = x(y−1s · y). This solves the case u1 = y±1 and themirror argument solves the case uk = x±1.

We have left to show the identity for u1 = z±1 and uk = z±1. It will sufficeto consider the case u1 = uk = z and the case u1 = z and uk = z−1. We putw = l(u2, . . . uk−1).

For the first case we will show x(zwz) · y = x · (zwz)y which is the same as[Lx, Ry](zwz) = zwz. Due to the induction assumption, we have [Lx, Ry](w) =w. We also have that [Lx, Ry](1) = 1, so due to Lemma 3.2 and Lemma 3.5, wehave that x(zwz) · y = x · (zwz)y.

16

For the second case we will show that x(zwz−1) · y = x · (zwz−1)y, i.e. thatα(z−1) = z−1 where α = L−1w L−1z [Ry, Lx]LzLw. By the induction assumption,we have x(zw · y) = (x · zw)y which is the same as α(1) = 1. We also haveα(z) = z from x(zwz) · y = x · (zwz)y. Then by Lemmas 3.1 and 3.5 we haveproven the identity in the second case.

What remains now is to prove the second equality of the proposition, still us-ing induction. We have l(u0, . . . , uk+1) = u0l(u1, . . . , uk+1) = u0r(u1, . . . , uk+1) =u0 · r(u1, . . . , uk)uk+1 which by the induction assumption and the proven firstequality of the proposition is equal to u0l(u1, . . . , uk)·uk+1 = l(u0, u1, . . . , uk)uk+1 =r(u0, . . . , uk)uk+1 = r(u0, . . . , uk+1).

Moufang’s Theorem. Let x,y and z be elements of a Moufang loop M. Ifx · yz = xy · z, then x, y and z generate a subgroup of M.

Proof. This follows from Lemma 3.3 and Proposition 3.1, since assuming x·yz =xy ·z holds for any x, y, z ∈M , any multiplication and inverses of these elementswill also associate, thus a group is generated.

Corollary 3.1. Any Moufang loop M is di-associative, meaning that any twoelements a, b ∈M generate a group.

Proof. This follows from Moufang’s Theorem taking the generating set as X ={a, b, b}. We have a · bb = ab · b due to alternativity of Moufang loops.

Corollary 3.2. Any Moufang loop M is power-associative, meaning that anyelement a ∈M generate a group.

Proof. Taking X = {a, a, a} as generating set, the corollary follows from Mo-ufang’s Theorem.

Moufang’s theorem in itself is of course of large importance, but in theremainder of this thesis, and especially in Chapter 5, Corollary 3.1 will have themost impact in the Moufang loops we study.

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Chapter 4

Moufang loops of smallorder

Since all groups are Moufang loops, the right term of the subject of this thesisis non-associative Moufang loops. However, for simplicity, we will sometimesuse Moufang loops even when we mean non-associative Moufang loops, and themeaning will be evident from the context.

This Chapter follows the outline of Chein [8] and additional clarificationshave been added when necessary.

As stated in the introduction, the purpose of this thesis is to get a generalunderstanding of Moufang loops. A good way to do this is to study Moufangloops of small order. It was earlier stated that the smallest Moufang loop is oforder 12. Table 4.1 [7] shows the number of non-associative Moufang loops oforder ≤ 32 and so a good limitation in this thesis is to study the Moufang loopsof order ≤ 31.

Table 4.1: Number π(n) of non-associative Moufang loops of order n

Order n π(n)12 116 520 124 528 132 60

In order to study Moufang loops of order ≤ 31 then it will be obvious thatit is crucial with an understanding of non-cyclic groups of order ≤ 15. We showin Table 4.2 these groups together with the number of elements in the minimalset of generators as well as the order of the generators in these sets.

In Table 4.2, we have the following notation:

18

• Cn is the cyclic group of order n

• V4 is Klein’s four group, sometimes noted C2 × C2

• Dn is the dihedral group of order 2n

• Sn is the symmetric group on n symbols

• Q is the group of units in the quaternions

• An is the alternating group on n symbols

• G12 is the remaining non-abelian group of order 12

Table 4.2: Noncyclic groups and order of elements in minimal set S

Group Order |S| Order of generators in S

V4 4 2 2 and 2

S3 = D3 6 2 2 and 3, or 2 and 2

C4 × C2 8 2 2 and 4, or 4 and 4

C2 × C2 × C2 8 3 2, 2 and 2

Q 8 2 4 and 4

D4 8 2 2 and 4, or 2 and 2

C3 × C3 9 2 3 and 3

D5 10 2 2 and 5, or 2 and 2

C6 × C2 12 2 2 and 6, or 6 and 6

A4 12 2 2 and 3, or 3 and 3

D6 12 2 2 and 6, or 2 and 2

G12 12 2 3 and 4, 4 and 4, or 6 and 4

D7 14 2 2 and 7, or 2 and 2

Lemma 4.1. If H is a subloop of a finite Moufang loop M, and if x ∈ M andx /∈ H, let d be the smallest divisor larger than 1 of |x| such that |x|/d dividesH. Then |〈H,x〉| ≥ d · |H|.

Proof. We consider all elements of the form hxi where h ∈ H and 0 ≤ i < d. Ifh1x

i = h2xj for i ≤ j we have h1 = (h2x

j)x−i = h2(xj−i), so xj−i = h−12 h1 ∈H. I we put r = j − i, then 0 ≤ r < d and xr ∈ H. If GCD(r, |x|) = t,then there exist integers u, v such that ru + |x|v = t. Thus, xt = xru+|x|v =(xr)u(x|x|)v = (xr)u ∈ H which means that |xt| divides |H|. We also have thatt divides |x|, so |xt| = |x|/t and thus t is a divisor of |x| such that |x|/t divides

19

|H|. However, given that r 6= 0 we have t ≤ r and we also know that r < d. Sosince d is chosen as the smallest divisor > 1 of |x| such that |x|/d divides H weeither have t = 1 or t = 0 and then r is in fact equal to 0 since t = GCD(r, |x|).But if t = 1 then x ∈ H, contrary to the assumption, so we have r = 0 andthus whenever h1x

i = h2xj we have i = j and h1 = h2. So the elements hxi,

h ∈ H, 0 ≤ i < d are distinct and there are |H| · d of them. We have of coursehxi ∈ 〈H,x〉, but these elements are not necessarily the only ones in 〈H,x〉 andso the lemma is proven.

This lemma might seem difficult to grasp, but Corollary 4.3 provides a goodexample on how it can be used in practice.

Corollary 4.1. If |〈H,x〉| = d · |H|, then every element of 〈H,x〉 may beuniquely expressed in the form hxi, h ∈ H, 0 ≤ i < d.

Proof. Since the elements hxi are distinct and since there are exactly d · |H| =|〈H,x〉| of them, they are all the elements in 〈H,x〉.

Corollary 4.2. If H and x are as in Lemma 4.1 and |x| is prime, then|〈H,x〉| ≥ |x||H|.

Proof. This should be obvious, since d = |x|.

Corollary 4.3. If a nonassociative Moufang loop M of order ≤ 31 containsan element z of order > 3, and if x, y ∈ M , y /∈ 〈z〉 and x /∈ 〈y, z〉, thenM = 〈x, y, z〉.

Proof. No matter the order of the elements x and y, their order’s smallestdivisors larger than 1 is at least 2. Thus, by Lemma 4.1,

|〈x, y, z〉| ≥ 2|〈y, z〉| ≥ 4|z| ≥ 16.

If M 6= 〈x, y, z〉, then there exists w ∈ M such that w /∈ 〈x, y, z〉. But then|M | ≥ |〈w, x, y, z〉| ≥ 2 · |〈x, y, z〉| ≥ 32 and we have a contradiction.

Corollary 4.3 is very important for our search in Moufang loops of order≤ 31.By di-associativity of Moufang loops, we know that non-associative Moufangloops cannot be generated by two elements. So we now know that we arelooking for loops generated by three elements or, if the loop is to have a minimalgenerating set of four elements or more, every element must have order ≤ 3.

Proposition 4.1. A Moufang loop of order p3, p prime, is a group.

This proposition will not be proven here, since the proof uses technicalitieswe have not gone through in this thesis. We state it here mainly because of oneof the cases in the Appendix uses its result, as well as its interesting Corollary:

Corollary 4.4. A Moufang loop M of order p or p2, p prime, is a group.

20

Proof. If |M | = p then |M × Cp2 | = p3 and if |M | = p2 then |M × Cp| =p3. Either way, M is isomorphic to a subloop of the direct product which, byProposition 4.1 is a group. Any subloop of a group is associative, so M is agroup.

In the search for Moufang loops M of order ≤ 31 we will consider all casesof the possible order of the elements in minimal sets of generators of M . Westart with the important case in which every set of generators of M contains anelement of order 2.

Theorem 4.1. If M is a nonassociative Moufang loop for which every minimalset of generators contains an element of order 2, then there exists a nonabeliangroup G, and an element x of order 2 in M, such that each element of M maybe uniquely expressed in the form gxα, where g ∈ G, α ∈ {0, 1}, and the productof two elements of M is given by

(g1xδ)(g2x

ε) = (gν1gµ2 )νxδ+ε (4.1)

where ν = (−1)ε and µ = (−1)ε+δ.Conversely, given any nonabelian group G, the loop M constructed as indi-

cated above is a nonassociative Moufang loop.

Proof. Let {x, u1, . . . , un} be a minimal set of generators for M containing thefewest possible elements of order 2. We can assume that x is of order 2. We letG = 〈u1, . . . , un〉. If g ∈ G then {gx, u1, . . . , un} generate M. Now, by the wayx, u1, . . . , un were chosen, gx must be of order 2. Hence, (gx)2 = gxgx = 1, so

gx = x−1g−1 = xg−1 (4.2)

For the sake of wanting to multiply any elements of the form gixε, we will

first study the multiplication g1(xεg2), ε ∈ {0, 1}. We have, due to the inverseproperty and a Moufang identity:

g1(xεg2) = (g2(g−12 g1))(xεg2) = (g2((g−12 g1)xε))g2

If ε = 0, then this is g1g2. If ε = 1, then {g2((g−12 g1)x), u1, . . . , un} is a gen-erating set for M and for the same reason as above, (g2((g−12 g1)x))2 = 1 andg2((g−12 g1)x) = (g2((g−12 g1)x))−1, meaning the element is its own inverse. Sim-ilarly, {(g−12 g1)x, u1, . . . , un} is a generating set such that ((g−12 g1)x)2 = 1 andthus (g−12 g1)x = ((g−12 g1)x)−1. Hence,

g1(xg2) = (g2((g−12 g1)x))g2 = (g2((g−12 g1)x))−1g2 = (((g−12 g1)x)−1g−12 )g2

= ((g−12 g1)x)−1 = x−1(g−12 g1)−1 = (g−12 g1)x = (g−11 g2)−1x

So we see that regardless of the value of ε, we have

g1(xεg2) = (gν1g2)νxε (4.3)

where ν = (−1)ε.

21

We can now multiply any elements in M . By the inverse property and aMoufang identity, we see that

(g1xδ)(g2x

ε) = (((g1xδ)(g2x

ε)xδ)x−δ = (g1(xδg2xε+δ))x−δ

Now if ε and δ are such that xε+δ = x then, due to (4.2), xδg2xε+δ = xδx−ε−δg−12 =

xεg−12 . If however, xε+δ = x0 = x2 = 1, then ε = δ and xδg2xε+δ = xδg2 = xεg2.

So we have

(g1xδ)(g2x

ε) = (g1(xεgµ2 ))x−δ

where µ = (−1)ε+δ. Hence, by (4.3),

(g1xδ)(g2x

ε) = ((gν1gµ2 )νxε)x−δ = (gν1g

µ2 )νxε+δ

We can conclude that the product of two elements of the form gxα, g ∈ G, isagain of that form. So the elements of the form gxα form a subloop of M . Butsince all elements x, u1, . . . , un can be expressed in this form, this subloop is allof M .

Every element has a unique representation in this form, since if g1xα = g2x

β

where α, β ∈ {0, 1}, then xα−β = g−11 g2 ∈ G. However, u1 . . . , un cannotgenerate M , so x /∈ G and thus α − β = 0 and α = β. Therefore, g1 = g2 andthe expression is unique.

We now need to show that elements of the form gxα, under the operationdefined by (4.1), form a nonassociative Moufang loop if and only if G is anonabelian group. We first clarify the product of two elements (g1x

δ)(g2xε) for

the three different cases when (δ, ε) 6= (0, 0).

(g1x)(g2x) = g−12 g1

(g1x)g2 = (g1g−12 )x

g1(g2x) = (g2g1)x

Assume that M is Moufang, and let a = g1, b = g2 and c = g3x. We knowthat (ab)(ca) = (a(bc))a and we have that

(ab)(ca) = (g1g2)((g3x)g1) = (g1g2)((g3g−11 )x) = ((g3g

−11 )(g1g2))x

And also

(a(bc))a = (g1(g2(g3x)))g1 = (g1((g3g2)x))g1

= (((g3g2)g1)x)g1 = (((g3g2)g1)g−11 )x = (g3g2)x

So (g3g−11 )(g1g2) = (g3g2) and ((g3g

−11 )(g1g2))g−11 = (g3g2)g−11 . Thus, due to a

Moufang identity and the inverse property

g3(g−11 (g1g2)g−11 ) = g3(g2g−11 ) = (g3g2)g−11

22

for any g1, g2, g3 ∈ G. Hence, when M is Moufang, G is an associative subloop ofM and therefore a group. To see when M is associative, let a = g1x

α, b = g2xβ

and c = g3xγ . Then

(ab)c = ((gν11 gµ1

2 )ν1xα+β)(g3xγ) = ((gν11 g

µ1

2 )ν1ν2gµ2

3 )ν2xα+β+γ ,

where ν1 = (−1)β , µ1 = (−1)α+β , ν2 = (−1)γ , µ2 = (−1)α+β+γ . We also have

a(bc) = (g1xα)((gν32 g

µ3

3 )ν3xβ+γ) = (gν41 (gν32 gµ3

3 )ν3µ4)ν4xα+β+γ ,

where ν3 = (−1)γ = ν2, µ3 = (−1)β+γ , ν4 = (−1)β+γ = µ3, µ4 = (−1)α+β+γ .So M is associative if and only if

((gν11 gµ1

2 )ν1ν2gµ2

3 )ν2 = (gµ3

1 (gν22 gµ3

3 )ν2µ2)µ3

Taking α = β = 0, γ = 1, g3 = 1 we get ((g1g2)−1)−1 = (g−11 (g−12 ))−1, whichbecomes g1g2 = g2g1. So if M is associative then G is abelian. Conversely, if Gis abelian, then

((gν11 gµ1

2 )ν1ν2gµ2

3 )ν1 = (gν11 gµ1

2 )ν1gµ2ν23 = g1g

µ1ν12 gµ2ν2

3 ,

using that ν1ν1 = 1. We also have

(gµ3

1 (gν22 gµ3

3 )ν2µ2)µ3 = g1gµ2µ3

2 gν2µ2

3 ,

using that µ3µ3 = ν2ν2 = 1. We have that µ1ν1 = (−1)α = µ2µ3 so (ab)c =a(bc) and M is associative.

To finish the proof, we need to show that if G is a group then M is Moufang.We must check the Moufang identity. We put a = g1x

α, b = g2xβ , c = g3x

γ .Then

(ab)(ca) = ((gν11 gµ1

2 )ν1xα+β)((gν23 gµ2

1 )ν2xα+γ) = ((gν11 gµ1

2 )ν1ν3(gν23 gµ2

1 )ν2µ3)ν3xβ+γ

and

(a(bc))a = ((g1xα)((gν42 g

µ4

3 )ν4xβ+γ))(g1xα)

= ((gν51 (gν42 gµ4

3 )ν4µ5)ν5xα+β+γ)(g1xα)

= ((gν51 (gν42 gµ4

3 )ν4µ5)ν5ν6gµ6

1 )ν6xβ+γ

where

µ1 = (−1)α+β , ν1 = (−1)β

µ2 = (−1)α+γ , ν2 = (−1)α

µ3 = (−1)β+γ , ν3 = (−1)α+γ = µ2

µ4 = (−1)β+γ = µ3, ν4 = (−1)γ

µ5 = (−1)α+β+γ , ν5 = (−1)β+γ = µ3

µ6 = (−1)β+γ = µ3, ν6 = (−1)α = ν2

23

So we need

((gν11 gµ1

2 )ν1ν3(gν23 gµ2

1 )ν2µ3)ν3 = ((gν51 (gν42 gµ4

3 )ν4µ5)ν5ν6gµ6

1 )ν6

for all cases of the values of α, β and γ. These eight cases are shown in Table4.3.

Table 4.3: Test of Moufang identity in M2n(G, 2)

α β γ (ab)(ca) (a(bc))a

0 0 1 ((g1g2)−1(g3g

−11 )−1)−1 = g3g2 (g−1

1 (g−12 g−1

3 ))−1g−11 = g3g2

0 1 0 (g−11 g−1

2 )−1(g3g1)−1 = g2g

−13 (g−1

1 (g2g−13 )−1)−1g−1

1 = g2g−13

0 1 1 ((g−11 g−1

2 )(g3g−11 ))−1 = g1g

−13 g2g1 (g1(g

−12 g3)

−1)g1 = g1g−13 g2g1

1 0 0 ((g1g−12 )−1(g−1

3 g−11 )−1)−1 = g−1

3 g−12 ((g1(g2g3)

−1)−1g1)−1 = g−1

3 g−12

1 0 1 ((g1g−12 )(g−1

3 g1)) = g1g−12 g−1

3 g1 ((g−11 (g−1

2 g−13 )−1)g−1

1 )−1 = g1g−12 g−1

3 g1

1 1 0 ((g−11 g2)(g

−13 g−1

1 ))−1 = g1g3g−12 g1 (g−1

1 (g2g−13 )g−1

1 )−1 = g1g3g−12 g1

1 1 1 (g−11 g2)

−1(g−13 g1)

−1 = g−12 g3 ((g1(g

−12 g3))

−1g1)−1 = g−1

2 g3

0 0 0 (g1g2)(g3g1) = g1g2g3g1 (g1(g2g3))g1 = g1g2g3g1

Thus M is Moufang and the proof is complete.

Theorem 4.1 not only covers every case of where each minimal generating setof a Moufang loop contains an element of order 2. It also gives rise to a methodfor finding Moufang loops of order 2n, where n is the order of a non-abeliangroup G. This class of Moufang loops is very important and a loop constructedin this way is denoted M2n(G, 2).

By the help of Table 4.2, we can now state the following Corollary.

Corollary 4.5. The only nonassociative Moufang loops of order ≤ 31 in whichevery minimal set of generators contains an element of order 2 are

M12(S3, 2) M24(A4, 2)

M16(Q, 2) M24(D6, 2)

M16(D4, 2) M24(G12, 2)

M20(D5, 2) M28(D7, 2)

We can now restrict our attention to Moufang loops where not all minimalsets of generators contain element of order 2. So, for every yz ∈M , there existsan x /∈ 〈y, z〉 such that |x| > 2.

Proposition 4.2. If a nonassociative Moufang loop M contains an element xsuch that |x| ≥ 8, then |M | > 31.

Proof. There must exist y, z ∈M such that y /∈ 〈x〉 and z /∈ 〈x, y〉. By Lemma4.1,

|M | ≥ |〈x, y, z〉| ≥ 2|〈x, y〉| ≥ 4|〈x〉| ≥ 32.

24

Proposition 4.3. The only nonassociative Moufang loops M of order ≤ 31which contains an element z of order 5 or 7 are M20(D5, 2) and M28(D7, 2).

Proof. If every minimal set of generators of M contains an element of order2, then the result follows from Corollary 4.5. Assume now that there exists aminimal set S of generators each of which is of order greater than 2.

Let a, b ∈ S. We note that 〈a, b〉 is not cyclic, since then S would notbe minimal. Since S is not associative, M 6= 〈a, b〉, and so, by Lemma 4.1,〈a, b〉 ≤ 15. If z ∈ 〈a, b〉, a survey of Table 4.2 implies that 〈a, b〉 = D5 orD7. But then either a or b would be of order 2, contrary to assumption. Soz /∈ 〈a, b〉. By Corollary 4.2, M = 〈z, a, b〉 and 〈a, b〉 ≤ 6. But then either aor b is of order 2, again contrary to assumption. We can thus conclude thatthe assumption that S does not contain an element of order 2 is false and theproposition follows.

The goal now is to find all non-associative Moufang loops of order less than32 where all elements in a minimal generating set are of order greater than 2.This search is divided into three cases, depending on the order of the elementsin M :

Case 1. M contains an element of order 6.Case 2. M contains an element of order 4, but no element of larger order.Case 3. M contains no element of order greater than 3.Studying these cases, we will find all non-associative Moufang loops of order

less than 32, that are not listed in Corollary 4.5. This case analysis is howeververy technical and not very enjoyable, so the calculations have been put in theAppendix, but the results of these cases are here presented.

Case 1 gives rise to the loops M24(G12, C2×C4) and M24(G12, Q) representedby

M24(G12, C2 × C4) =〈x, y, z : x2 = y2 = z3, z6 = 1,

((xαyβ)zγ)((xσyτ )zρ) = (xα+σyβ+τ )zµγ+νρ,

where µ = (−1)σ+τ+στ and ν = (−1)ατ+βσ〉.

M24(G12, Q) =〈x, y, z : x2 = y2 = z3, z6 = 1,

((xαyβ)zγ)((xσyτ )zρ) = (xα+σyβ+τ )z3βσ+µγ+νρ,

where µ = (−1)σ+τ+στ and ν = (−1)ατ+βσ〉.

Case 2 gives rise to three loops, M16(C2×C4), M16(C2×C4, Q) and M16(Q),represented by

25

M16(C2 × C4) =〈x, y, z : x2 = y2 = z2; z4 = 1;

(xαyβ)zγ · (xσyτ )zρ = (xα+σyβ+τ )zµγ+νρ,

where µ = (−1)στ and ν = (−1)ατ+βσ〉

M16(C2 × C4, Q) =〈x, y, z : x2 = y2 = z2; z4 = 1;

((xαyβ)zγ)((xσyτ )zρ) = (xα+σyβ+τ )zµγ+νρ+2βσ,

where µ = (−1)στ and ν = (−1)ατ+βσ

M16(Q) =〈x, y, z : x2 = y2 = z2; z4 = 1;

((xαyβ)zγ)((xσyτ )zρ) = (xα+σyβ+τ )zµγ+νρ+2βσ,

where µ = (−1)σ+τ+στ and ν = (−1)ατ+βσ

Case 3 does not give rise to any non-associative Moufang loops.

26

Chapter 5

Visualization of Moufangloops of order 16

Many groups of small order have fairly simple multiplication rules, making iteasy to get a sense of the structure of them. But the non-associative Moufangloops presented in the last chapter have more complicated binary operations,and thus it would be of interest to provide some sort of visualization of the loops.Petr Vojtechovsky [6] presents graphs for the multiplication in M12(S3, 2) in away that visualize the multiplication between all of the elements in the loop.The graphs also illustrate the di-associativity of Moufang loops by showingwhat subgroups all pairs of elements generate. These graphs do not follow anycommon standard and will here be referred to as multiplication graphs.

Table 5.1 [8] shows non-associative Moufang loops of order 12 and 16 alongwith the subgroups generated by two or more elements, nucleusN and number ofelements of order 2,3 and 4. All loops have of course subgroups generated by oneelement, but how these are generated is obvious. As we present multiplicationgraphs for these loops, the subgroups will emerge from patterns in the graphs,so bare in mind this table as we continue.

Table 5.1: Moufang loops of order 12 and 16

Subgroups generated bytwo or more elements

Order ofelements

Moufang loop N 2 3 4

M12(S3, 2) S3, V4 1 9 2 -

M16(D4, 2) D4, C2 × C2 × C2, V4 C2 13 - 2

M16(Q, 2) Q, D4, V4 C2 9 - 6

M16(C2 × C4) C2 × C4, D4, V4 C2 9 - 6

M16(C2 × C4, Q) C2 × C4, Q, D4, V4 C2 5 - 10

M16(Q) Q C2 1 - 14

27

The Cayley tables of the loops we present here are located in the Appendixwith the name of the elements corresponding to how we name them in thischapter. We thus leave the notation of the elements in the loops used in Chapter4. Except for in the case of M12(S3, 2), the graphs of the loops in this chapterare constructed for this thesis by using the loops’ Cayley tables.

The multiplication graphs are constructed in the following way: M12(S3, 2)contains nine involutions (elements of order 2) and two elements of order 3. Theinvolutions are named x1, . . . x9 and the elements of order 3 are named y andy−1. The involutions are then put as vertices in a circle as shown in Figure5.1. Multiplication between two involutions are illustrated by an edge and ifthat edge is solid, then the product of those elements is the third vertex in thetriangle that the multiplication gives rise to. If two involutions are connected bya dotted line, then multiplication between these elements result in an elementof order 3. This multiplication we call clockwise positive, meaning that, forexample x1 · x4 = y and x4 · x1 = y−1.

Figure 5.1: Multiplication and generating sets in M12(S3, 2)

x1x2

x3

x4

x5x6

x7

x8

x9x1

x2

x3

x4

x5x6

x7

x8

x9x1

x2

x3

x4

x5x6

x7

x8

x9

y±1

x1x2

x3

x4

x5x6

x7

x8

x9

The subgroup structure of M12(S3, 2) is now evident. Any two elementsxi, xj that are connected by a solid edge generate V4. This subgroup then con-sists of xi, xj and the third involution in the unique triangle that multiplicationbetween xi and xj gives rise to, as well as the identity element e, not shown inthe graphs.

Any two elements xi, xj connected by a dotted edge generate S3. Thissubgroup then consists of xi, xj , the third involution in the dotted triangle thatmultiplication between xi and xj gives rise to, as well as e, y and y−1.

We now use this way of constructing multiplication graphs on the Moufangloops of order 16 found in the last chapter.

28

5.1 M16(D4, 2)

This Moufang loop consists of 13 involutions and two elements of order 4. How-ever, as seen in Table 5.1, the nucleus of M16(D4, 2) consists of two elements;the identity element and an element of order 2. We call this involution n and sothe involutions of M16(D4, 2) are {x1, . . . , x12, n}. We treat multiplication byn differently, so the vertices in the multiplication graph of M16(D4, 2) are onlyx1, . . . , x12. The elements are then ordered in such a way that the product ofinvolutions on the opposite side of each other is n. Graphs I-III in Figure 5.2shows multiplication between all involutions x1, . . . , x12. As before, a solid edgebetween two elements implies that the product of these elements is the elementin the last vertex of the unique triangle that the multiplication gives rise to.

Figure 5.2: Multiplication and generating sets in M16(D4, 2)

x1 x2

x3

x4

x5

x6x7x8

x9

x10

x11

x12I x1 x2

x3

x4

x5

x6x7x8

x9

x10

x11

x12II x1 x2

x3

x4

x5

x6x7x8

x9

x10

x11

x12III x1 x2

x3

x4

x5

x6x7x8

x9

x10

x11

x12

y±1

IV

All pairs of elements that are connected by a solid edge in graphs I-III inFigure 5.2 generate V4. The elements of V4 is then the elements in a triangleand the identity element e as in the first five graphs.

Again, we denote the two elements of order 4 by y and y−1. In graph IV inFigure 5.2, we see the multiplication of involutions such that the product is anelement of order 4. in fact, the multiplication is clockwise positive, so x1 ·x4 = yand x4 · x1 = y−1 as before. We thus see that for any xi, xj connected by adotted edge, 〈xi, xj〉 ∼= D4 and the elements in D4 are then all elements in thesquare where xi and xj are vertices, as well as y, y−1, e and n.

29

5.2 M16(Q, 2)

Looking at Table 5.1, we see that M16(Q, 2) have subgroups Q,D4 and V4. Howare these generated? We also see that it has nine involutions and six elementsof order 4. However, as in M16(D4, 2), we have two elements in the nucleus, eand the one we call n which again is an involution. The elements of order 4 aredenoted y1, y

−11 , y2, y

−12 , y3, y

−13 .

Figure 5.3: Multiplication and generating sets in M16(Q, 2)

x1x2

x3

x4x5

x6

x7

x8

y±11

I x1x2

x3

x4x5

x6

x7

x8

y±12

II x1x2

x3

x4x5

x6

x7

x8

y±13

III y1

y2

y3

y−11

y−12

y−13

IV

The multiplication graphs I-III in Figure 5.3 have involutions x1, . . . , x8 asvertices. We see that if two involutions are connected by a directed edge, thenthe product of these elements is one of the elements of order 4 in the center ofthe graph. Multiplication in the direction of the arrows is ”positive”, so that,for example, x1 ·x2 = y2 and x2 ·x1 = y−12 . As before, the product of involutionson the opposite side of each other is n.

Two involutions that are connected by a directed edge generate all fourinvolutions in the rectangle wherein they are vertices, as well as the elementsof order 4 in the center of that rectangle. They also generate e and n and so〈xi, xj〉 ∼= D4 where xi and xj are connected by a directed edge. Two involutionson the opposite side of each other generate V4.

In the previous loops, we have only two elements of order 3 or 4. Now thatwe have six elements of order 4, we show the multiplication of these elements ingraph IV of Figure 5.3. The graph shows six directed 3-cycles. Multiplication oftwo elements in the direction of the 3-cycle results in the third element of thatcycle. Note that all pairs of elements that are not on the opposite side of eachother are contained in two different 3-cycles, and that, for example y1 · y2 = y3but y2 · y1 = y−13 .

We have that any two elements in graph IV in Figure 5.3 that are notinverses generate all the other elements in the graph, as well as e and n (since(y±1i )2 ∈ {e, n} for any i). Thus, 〈y±1i , y±1j 〉 ∼= Q for any yi, yj such that

y±1i 6= y±1j or y±1i · y±1j 6= e.

30

5.3 M16(C2 × C4)

Figure 5.4:

x1

x2

x3

x4

x5

x6

The multiplication graphs we use to describe the mul-tiplication in Moufang loops in this thesis do not fol-low an exact standard. The graphs we have seen sofar could have looked different if the vertices was cho-sen to represent other elements in the loop or if theywere put in a different order. So far, it has howeverbeen a quite obvious choice of how to represent theloops. Turning now to the loop M16(C2×C4) we willsee that the choice of vertices are a bit different but

the underlying reason for this is to produce graphs that are most easily readand that show the symmetry of the loops in the clearest way.

We will now abandon the objective of trying to find multiplication rules forall of the elements in M16(C2 × C4), due to the more complicated structure ofthis loop. Instead, we focus on visualizing the di-associativity of the loop. First,Figure 5.4 shows six elements in M16(C2 × C4) that are involutions. These dohave a simple multiplication rule similar to the ones of involutions in the previousloops. The product of two elements joined by an edge is the third element inthat triangle. And as before, 〈xi, xj〉 ∼= V4 if xi and xj are connected by anedge.

Figure 5.5: Elements in M16(C2 × C4) generating D4 or C2 × C4

x1 x2

x3x4x5

x6

x′7x′8

y1

y2y−11

y−12

I x1 x2

x3x4x5

x6

x′7x′8

y1

y2y−11

y−12

II

y±13

x1 x2

x3x4x5

x6

x′7x′8

y1

y2y−11

y−12

III

x1 x2

x3x4x5

x6

x′7x′8

y1

y2y−11

y−12

IV y±13

x1 x2

x3x4x5

x6

x′7x′8

y1

y2y3

y4

V

Looking now at Figure 5.5, x1, . . . , x6 are kept as vertices in the graphs,and four elements of order 4 are added; y1, y

−11 , y2 and y−12 , as well as two

involutions, denoted x′7 and x′8. In graphs I and II, any two elements joined byan edge generate D4 where the elements are all of the elements that are verticesjoined by an edge, as well as e and n, where n again is the involution in thenucleus.

In graph III in Figure 5.5, we have that the product of two elements joinedby an edge is one of the elements in the middle. These elements y3 and y−13 arethe last two elements of order 4. Again, multiplication is clockwise positive and

31

elements connected by an edge generate D4.Lastly, we look at graphs IV and V in Figure 5.5, where in IV, elements

joined by a dotted edge generate C2 × C4 consisting of all the elements thatare vertices joined by an edge, as well as e and n. In graph V, the clockwiseproduct of elements joined by a dotted edge is y3, while the counter-clockwiseproduct is y−13 . These elements also generate C2 × C4.

32

5.4 M16(C2 × C4, Q)

As in M16(C2×C4), the graphs of this loop will mostly show which elements thatgenerate which subgroup. In the graphs, the vertices was chosen and ordered sothat the subgroup structure was most evident. The twelve vertices are the fourinvolutions x1, . . . , x4 and the eight elements of order 4, y±11 , y±12 , y±13 , y±14 .

Figure 5.6: Elements in M16(C2 × C4, Q) generating C2 × C4

x1

x2

x3

x4

y1

y2

y3

y4y−11

y−12

y−13

y−14

I x1

x2

x3

x4

y1

y2

y3

y4y−11

y−12

y−13

y−14

II x1

x2

x3

x4

y1

y2

y3

y4y−11

y−12

y−13

y−14

III x1

x2

x3

x4

y1

y2

y3

y4y−11

y−12

y−13

y−14

IV

Figure 5.7: Elements in M16(C2 × C4, Q) generating Q or D4

y±15

x1

x2

x3

x4

y1

y2

y3

y4y−11

y−12

y−13

y−14

I

y±15

x1

x2

x3

x4

y1

y2

y3

y4y−11

y−12

y−13

y−14

II

Figure 5.6 shows which elements that generate C2×C4. Any pair of elementsthat are connected by an edge generate all other elements connected by the edgesin that figure, as well as e and n. In Figure 5.7 we have the same multiplicationrule as seen before. Clockwise multiplication of two elements connected bya dotted or dashed edge results in y5, another element of order 4, while thecounter-clockwise product is y−15 .

Any pair of elements connected by a dotted line in graph I in Figure 5.7generate the group Q, while any pair of elements connected by a dashed line ingraph II generate D4.

It is not shown graphically, but any involutions xi, xj such that xi · xj = ngenerate V4.

33

5.5 M16(Q)

We have reached the last visualization of Moufang loops in this thesis, M16(Q)1.Table 5.1 shows that the only subgroup of M16(Q) generated by two elements isQ. We thus have that any two elements a, b ∈M16(Q) such that a 6= b, a 6= b−1

or a, b /∈ N generate Q. Figure 5.8 shows this fact graphically. All elements notin the nucleus is of order 4 so we denote them by y±11 , . . . , y±17 . The first graphshows that, for example, y1 · y4 = y7 and y4 · y1 = y−17 in the same way as seenpreviously.

Figure 5.8: Elements in M16(Q) generating Q

y1 y2

y3

y4

y5

y6y−11

y−12

y−13

y−14

y−15

y−16

y±17

y1 y2

y3

y4

y5

y6y−11

y−12

y−13

y−14

y−15

y−16

y1 y2

y3

y4

y5

y6y−11

y−12

y−13

y−14

y−15

y−16

y1 y2

y3

y4

y5

y6y−11

y−12

y−13

y−14

y−15

y−16

y1 y2

y3

y4

y5

y6y−11

y−12

y−13

y−14

y−15

y−16

1The non-associative Moufang loop M16(Q) is commonly known as the standard basis ofthe octonions [1].

34

Chapter 6

Appendix

6.1 Case studies of elements of different order in Moufang

loops

We will assume a familiarity with Table 4.2 when studying these cases.Case 1. M contains an element z of order 6. If x /∈ 〈z〉, then, since M

cannot be generated by only two elements, M contains an element y /∈ 〈x, z〉.By Lemma 4.1,

31 ≥ |M | ≥ |〈x, y, z〉| ≥ 2|〈x, z〉|,

so |〈x, z〉| ≤ 15. Since |z| = 6 we have that 〈x, z〉 is a group containing anelement of order 6. Studying Table 4.2, we see that we have 〈x, z〉 ∼= C6 × C2,D6 or G12.

We now consider several subcases.Case 1(a). There exist x and y both of order 6 in M , such that y /∈ 〈z〉

and x /∈ 〈y, z〉. In this case, 〈x, y〉 ∼= 〈x, z〉 ∼= 〈y, z〉 ∼= C2 × C6. A knowledge ofC2×C6 tells us that we can, independently of z, choose x and y such that x2 =y2 = z2, so we can assume that this equality holds. Now since 〈x, y〉 ∼= C2×C6,and x2 = y2 we have |xy| = 6. So 〈xy, z〉 ∼= C2×C6 and thus (xy) = z = z(xy).A Moufang identity now give us that

((xy)z)y = x(yzy) = (x(y2z) = xz3 = (xz)z2 = (xz)y2

so that

z(xy) = (xy)z = ((xz)y2)y−1 = (xz)y = (zx)y.

Thus M is a group and case 1(a) does not result in a non-associative Moufangloop.

Case 1(b). There exists y /∈ 〈z〉 such that |y| = 6, but for every x /∈ 〈y, z〉,|x| 6= 6. Since 〈x, z〉 ∼= C2 × C6, D6 or G12, |x| = 2 or 4. (A knowledge of thegroups in question tells us that |x| 6= 3). Now if |x| = 2 for each x /∈ 〈y, z〉, then

35

every minimal set of generators of M contains an element of order 2, resultingin a Moufang loop we have already found. Hence |x| = 4 for some x /∈ 〈y, z〉.〈y, z〉 ∼= C2 × C6, and we can again assume that y2 = z2. We m¡ust have

that 〈x, z〉 ∼= 〈x, y〉 ∼= G12, and thus x2 = z3 and x2 = y3 implying y3 = z3. Butthen y = z, contrary to assumption, so case 1(b) cannot occur.

Case 1(c). For each w /∈ 〈z〉, |w| < 6. Again, we have that |w| = 2 or 4, andsince not every minimal set of generators of M contains an element of order 2,there exist x, y ∈M such that |x| = |y| = 4, and, by Corollary 4.3, M = 〈x, y, z〉.From Table 4.2 we see that 〈x, z〉 ∼= 〈y, z〉 ∼= G12 and 〈x, y〉 ∼= C2 × C4, Q, orG12. However, if 〈x, y〉 ∼= G12, then there exists an element of order 6 which isnot in 〈z〉, contrary to assumption. So 〈x, y〉 ∼= C2 × C4 or Q. In either case,x2 = y2 = z3 and zx = xz−1 and zy = yz−1.

If 〈x, y〉 ∼= C2 × C4, then (xy)2 = 1, so 〈xy, z〉 ∼= C2 × C6 or D6. In theformer case |(xy)z| = 6, contradicting that all w /∈ 〈z〉 have order less than 6,so we must have 〈xy, z〉 ∼= D6 and thus z(xy) = (xy)z−1.

If instead 〈x, y〉 ∼= Q, then |xy| = 4 and so 〈xy, z〉 ∼= G12 and again z(xy) =(xy)z−1.

We now consider the multiplication in M of elements of the form (xαyβ)zγ .Since x2 = y2 = z3, we can assume that 0 ≤ α, β ≤ 1, 0 ≤ γ ≤ 5 so that, forexample,

(x3y)z = (xy3)z5 = ((xy)z3)z5 = (xy)z2.

Since x, y and z are on this form and M = 〈x, y, z〉 then if we can show thatthe product of two elements of this form is again of this form, every element ofM is expressible in this form. We have that |M | = |〈x, y, z〉| ≥ 2|〈y, z〉| = 24and there are at most 24 distinct element of this form, so every element is thenuniquely expressible in this form.

To investigate the multiplication of (xαyβ)zγ by (xσyτ )zρ, let u = xαyβ andv = xσyτ . If v 6= 1 (i.e. v = x, y or xy), then zv = vz−1 and thus

((uzγ)(vzρ))zγ = u(zγvzγ+ρ) =

{uz2γ+ρ if v = 1

u(vzρ) otherwise.

If v 6= 1 but u = 1, this is vzρ and if u = v 6= 1, we get u2zρ. If u2 6= 1 thenu2 = z3. Hence, if u 6= 1, v 6= 1, u 6= v, and θ = 0 if u2 = 1 and θ = 0 otherwise,we have

u(vzρ) = u(z−ρv) = u((zθ−ρu−2)v) = u((u−1zρ−θu−1)v)

= u(u−1(zρ−θ(u−1v))) = zρ−θ(u−1v) = (u−1v)zθ−ρ

= ((uz−θ)v)zθ−ρ = (uv)zρ.

36

Summarizing, we have that

(uzγ)(vzρ) =

uzγ+ρ if v = 1

vzρ−γ if v 6= 1, u = 1

u2zρ−γ if u = v 6= 1

(uv)z−ρ−γ otherwise

Now setting

µ =

{1 if v = 1

−1 otherwiseand ν =

{1 if u = 1, or v = 1, or u = v

−1 otherwise

we get (uzγ)(vzρ) = (uv)zµγ+νρ.Replacing u and v by their expressions in terms of x and y, we get

((xαyβ)zγ)((xσyτ )zρ) = (xαyβxσyτ )zµγ+νρ (6.1)

where

µ =

{1 if σ = τ = 0

−1 otherwise= (−1)σ+τ+στ

and

ν =

{1 if α = β = 0, or σ = τ = 0, or α = σ, β = τ

−1 otherwise= (−1)ατ+βσ

We need now to consider the two different cases 〈x, y〉 ∼= C2×C4 and 〈x, y〉 ∼=Q.

Case 1(c1). If 〈x, y〉 ∼= C2 × C4, then xy = yx and (6.1) becomes

((xαyβ)zγ)((xσyτ )zρ) = (xα+σyβ+τ )zµγ+νρ

where µ and ν are defined as above. This product is again of the form (xαyβ)zγ

and thus every element of M is uniquely expressible in this form. So the achievedMoufang loop has the following representation:

M = 〈x, y, z : x2 = y2 = z3, z6 = 1, ((xαyβ)zγ)((xσyτ )zρ) = (xα+σyβ+τ )zµγ+νρ,

where µ and ν are defined as above〉.

It is now needed to check that this loop is in fact Moufang, but this is, asseen before, very tedious and will be omitted. We can however easily see thatM is not a group since x(yz) = (xy)z−1 = (xy)z5 6= (xy)z.

This Moufang loop found in case 1(c1) is noted as M24(G12, C2 × C4).Case 1(c2). We have here that 〈x, y〉 ∼= Q, so that yx = xy3 = (xy)y2 =

(xy)z3, and hence, xαyβxσyτ = (xα+σyβ+τ )z3βσ.

37

So (6.1) here becomes

((xαyβ)zγ)((xσyτ )zρ) = (xα+σyβ+τ )z3βσ+µγ+νρ,

where µ and ν are defined as above.Case 1(c2) thus gives rise to a Moufang loop with the presentation

M = 〈x, y, z : x2 = y2 = z3, z6 = 1, ((xαyβ)zγ)((xσyτ )zρ)

= (xα+σyβ+τ )z3βσ+µγ+νρ, where µ and ν are defined as above〉.

Again, it is tedious to show that M is Moufang, but for the same reasonas in Case 1(c1), M is not a group. The Moufang loop found here is denotedM24(G12, Q).

Case 2. We now assume that M contains an element x of order 4, butno element of higher order. Assume also that M also contains an element y,which means that, from di-associativity, 〈x, y〉 is a group. If now y is of order3, then we know from Table 4.2 that either |〈x, y〉| > 15 or 〈x, y〉 ∼= C12 orG12. If |〈x, y〉| > 15, then |M | > 31 contrary to assumption, and if 〈x, y〉 ∼= C12

or G12, then M contains an element of order greater than 4, also contraryto assumption. So M contains no element of order 3, and thus, apart fromthe identity element, M contains only elements of order 2 or 4. Since not everyminimal set of generators contain an element of order 2, there exist, by Corollary4.3, x, y, z ∈M , all of order 4 such that 〈x, y, z〉 = M .

We use now a result from Glauberman, Wright [5] implying that if eachelement of a Moufang loop is of order a power of 2, we have |M | = 2k for somek. Since |M | ≤ 31 we have |M | ≤ 16 and thus

16 ≥ |M | = |〈x, y, z〉| ≥ 2|〈x, y〉| ≥ 4|x| = 16

and so |M | = 16 and |〈x, y〉| = 8.From Table 4.2, we thus have 〈x, y〉 ∼= C2 × C4 or Q. Either way, x2 =

y2, and, by analogous reasoning x2 = y2 = z2. Every element in M can beexpressed as (xαyβ)zγ where α, β ∈ {0, 1} and 0 ≤ γ ≤ 3. Since |M | = 16and there are 16 elements of this form and if (xa1yb1)zc1 = (xa2yb2)zc3 thenzc1−c2 = (xa2yb2)(xa1yb1)−1 ∈ 〈x, y〉 and thus c1 = c2, z

c1 = zc2 and alsoxa1 = xa2 , yb1 = yb2 , each element of this form is therefore distinct. We neednow determine how to multiply two elements of this form.

For any w ∈ M , by Table 4.2 we have 〈w, x〉 ∼= C2 × C4, D4, Q or C4. Inany of these cases, wx2 = x2w, so x2 is in the Moufang center of M and since(x2)2 = x4 = 1, x2 is in the center of M , again using Glauberman, Wright[5]. We also have, for any w ∈ M , either w2 = x2 or w2 = 1, since otherwise|w| /∈ {2, 4}

As mentioned, we have 〈x, y〉 ∼= C2 × C4 or Q and this also holds for 〈x, z〉and 〈y, z〉. There are thus four cases that need to be considered:

Case 2(a) 〈x, y〉 ∼= 〈x, z〉 ∼= 〈y, z〉 ∼= C2 × C4

Case 2(b) 〈x, y〉 ∼= Q, 〈x, z〉 ∼= 〈y, z〉 ∼= C2 × C4

Case 2(c) 〈x, y〉 ∼= C2 × C4, 〈x, z〉 ∼= 〈y, z〉 ∼= Q

38

Case 2(d) 〈x, y〉 ∼= 〈x, z〉 ∼= 〈y, z〉 ∼= QIn both Case 2(a) and 2(b), we have

z(xy) = ((z(xy))z)z3 = ((zx)(yz))z3 = ((xz)(yz))z · z2 = (x(zyz2))z2

= (x(yz3))z2 = x(yz)

using that M is Moufang, that z2 is in the center of M and that C2 × C4 isabelian. Now we know, by Moufang’s theorem, that x(yz) 6= (xy)z since M isnot a group. Thus, z(xy) 6= (xy)z.

In Case 2(a), |xy| = 2, so we have that 〈xy, z〉 ∼= C2 × C4 or D4. But sincez(xy) 6= (xy)z we must have 〈xy, z〉 ∼= D4 and thus z(xy) = (xy)z−1.

Similarly, in case 2(b), |xy| = 4 so 〈xy, z〉 ∼= C2 × C4 or Q. Again, z(xy) 6=(xy)z so 〈xy, z〉 ∼= Q and thus z(xy) = (xy)z−1 here as well.

However, if any of x or y is raised to a power of 2, then, for example,z(x2y) = (x2y)z, since x2 = z2 and z2 both commutes and associates with allother elements in M . Also zy = yz and zx = xz. Thus in cases 2(a) and 2(b),

zη(xδyε) =

{(xδyε)z−η if δ ≡ ε ≡ 1 (mod 2)

(xδyε)zη otherwise

We now define µ, φ, ψ and ν respectively by

µ = (−1)στ , φ = (−1)αβ , ψ = (−1)(α+σ)(β+τ), ν = (−1)ατ+βσ

Since both x2 and y2 are in the center of M , in cases 2(a) and 2(b) we have

(xαyβ)((xσyτ )zρ) = (xαyβ)(zµρ(xσyτ )) (6.2)

In case 2(a), x and y commute, so this is equal to

(xαyβ)(zµρ((xαyβ)(xσ−αyτ−β))) = (((xαyβ)zµρ)(xαyβ))(xσ−αyτ−β)

= (zφµρ(x2αy2β))(xσ−αyτ−β) = zφµρ(xα+σyβ+τ ) = (xα+σyβ+τ )zψφµρ

= xα+σyβ+τzνρ.

due to a Moufang identity, alternativity, the fact that x2αy2β is in the centerand that ψφρ = (−1)ατ+βσ = ν. We can now conclude:

((xαyβ)zγ)((xσyτ )zρ) = ((xαyβ)(zγ(xσyτ )zρ+γ))z−γ

= ((xαyβ)((xσyτ )zρ+γ+µγ))z−γ = (xα+σyβ+τ )zν(ρ+γ+µγ)−γ = (xα+σyβ+τ )zµγ+νρ

So the loop in case 2(a) is

M = 〈x, y, z : x2 = y2 = z2; z4 = 1; (xαyβ)zγ · (xσyτ )zρ = (xα+σyβ+τ )zµγ+νρ,

where µ = (−1)στ and ν = (−1)ατ+βσ〉

We again omit the check that M is a non-associative Moufang loop. This loopis commonly denoted M16(C2 × C4).

39

In case 2(b), we have yx = xy3 = (xy)y2 = (xy)z2, and z2 is in the centerof M , so (6.2) becomes

(xαyβ)((xστ)zρ) = (xαyβ)(zµρ((xαyβ)((xσ−αyτ−β)z2β(σ−α))))

= ((xαyβ)zµρ(xαyβ))((xσ−αyτ−β)z2β(σ−α))

= (zφµρ(x2αy2βz2αβ))((xσ−αyτ−β)z2β(σ−α))

= zφµρ(xα+σyβ+τ )z2β(σ−α)+2αβ+4β(σ−α) = zφµρ(xα+σyβ+τ )z2βσ

= (xα+σyβ+τ )zψφµρ+2βσ = (xα+σyβ+τ )zµγ+νρ+2βσ

We skip the calculation of ((xαyβ)zγ)((xσyτ )zρ) but this multiplication isstraightforward and eventually results in

((xαyβ)zγ)((xσyτ )zρ) = (xα+σyβ+τ )zµγ+νρ+2βσ

So we find that the loop in case 2(b) is given by

M = 〈x, y, z : x2 = y2 = z2; z4 = 1; ((xαyβ)zγ)((xσyτ )zρ) = (xα+σyβ+τ )zµγ+νρ+2βσ,

where µ = (−1)στ and ν = (−1)ατ+βσ

and this is denoted M16(C2 × C4, Q).In Case 2(c), we have 〈x, z〉 ∼= Q, so |xz| = 4 and 〈x, xz〉 ∼= Q. As in 2(a),

〈xy, z〉 ∼= C2 × C4 or D4, and thus

z(xy) = (z(xy)z)z3 = ((zx)(yz))z3 = ((xz3)(yz))z3 = x(z3y) = x(yz) 6= (xy)z

So we have z(xy) = (xy)z3. Now consider 〈y, xz〉. Here we have

y(xz) = zz3(y(z3x)) = ((z3yz3)x) = z(yx) = z(xy) = (xy)z3

Also

(xz)y = ((xz)y)z · z3 = (x(zyz))z3 = (xy)z3

so y(xz) = (xz)y which means 〈y, xz〉 ∼= C2 × C4. Choosing x, xz, y as our setof generators for a loop M , we have 〈x, xz〉 ∼= Q, 〈x, y〉 ∼= 〈xz, y〉 ∼= C2×C4. Wesee that Case 2(c) in fact results in the same loop as in Case 2(b).

In Case 2(d) we again find that z(xy) = (xy)z−1. However, this time

zη(xδyε) =

{(xδyε)zη if δ ≡ ε ≡ 2 (mod 2)

(xδyε)z−η otherwise

Following the manner of the previous cases, we eventually find that

(xαyβ)zγ · (xσyτ )zρ = (xα+σyβ+τ )zµγ+νρ+2βσ

where ν = (−1)ατ+βσ as before, but now µ = (−1)σ+τ+στ .

40

The loop in case 2(d) is thus given by

M = 〈x, y, z : x2 = y2 = z2; z4 = 1; ((xαyβ)zγ)((xσyτ )zρ) = (xα+σyβ+τ )zµγ+νρ+2βσ,

where µ = (−1)σ+τ+στ and ν = (−1)ατ+βσ

and is denoted M16(Q). It will be omitted to show that it is in fact a non-associative Moufang loop.

Case 3. M contains no element of order greater than 3. Since we are alsoassuming that not all minimal sets of generators of M contain an element oforder 2, M must contain a minimal set of generators such that each elementin the set is of order 3. From Lemma 4.1 and its corollaries, such a set mustcontain exactly 3 elements, here called x, y and z.

Let K = {g : g ∈M, |g| = 3} and g1, g2 ∈ K. By Table 4.2, either 〈g1, g2〉 ∼=A4 or C3×C3. We assume that x /∈ 〈g1, g2〉. If 〈g1, g2〉 ∼= A4, then, by Corollary4.2, |〈g1, g2, x〉| ≥ |A4||x| = 36, contrary to assumption. So 〈g1, g2〉 ∼= C3 × C3

and whichever elements g1 and g2 are, we have |g1g2| = 3. So K has exponent3, meaning that for any element a ∈ K, a3 = e, and since x, y, z ∈ K we haveK = M . Now, by Theorem 10.1 in [2], M is of order 33 and by

So case 3 gives rise to no nonassociative Moufang loops.

41

6.2 Cayley tables for Moufang loops of order 16

Table 6.1: Cayley table of M16(D4, 2)e y n y−1 x12 x3 x6 x9 x1 x10 x7 x4 x2 x11 x8 x5

y n y−1 e x9 x12 x3 x6 x10 x7 x4 x1 x11 x8 x5 x2

n y−1 e y x6 x9 x12 x3 x7 x4 x1 x10 x8 x5 x2 x11

y−1 e y n x3 x6 x9 x12 x4 x1 x10 x7 x5 x2 x11 x8

x12 x3 x6 x9 e y n y−1 x2 x5 x8 x11 x1 x4 x7 x10

x3 x6 x9 x12 y−1 e y n x11 x2 x5 x8 x10 x1 x4 x7

x6 x9 x12 x3 n y−1 e y x8 x11 x2 x5 x7 x10 x1 x4

x9 x12 x3 x6 y n y−1 e x5 x8 x11 x2 x4 x7 x10 x1

x1 x4 x7 x10 x2 x11 x8 x5 e y−1 n y x12 x3 x6 x9

x10 x1 x4 x7 x5 x2 x11 x8 y e y−1 n x3 x6 x9 x12

x7 x10 x1 x4 x8 x5 x2 x11 n y e y−1 x6 x9 x12 x3

x4 x7 x10 x1 x11 x8 x5 x2 y−1 n y e x9 x12 x3 x6

x2 x5 x8 x11 x1 x10 x7 x4 x12 x3 x6 x9 e y−1 n yx11 x2 x5 x8 x4 x1 x10 x7 x3 x6 x9 x12 y e y−1 nx8 x11 x2 x5 x7 x4 x1 x10 x6 x9 x12 x3 n y e y−1

x5 x8 x11 x2 x10 x7 x4 x1 x9 x12 x3 x6 y−1 n y e

Table 6.2: Cayley table of M16(Q, 2)e y1 n y−1

1 y2 y3 y−12 y−1

3 x1 x3 x5 x7 x2 x4 x6 x8

y1 n y−11 e y3 y−1

2 y−13 y2 x7 x1 x3 x5 x4 x6 x8 x2

n y−11 e y1 y−1

2 y−13 y2 y3 x5 x7 x1 x3 x6 x8 x2 x4

y−11 e y1 n y−1

3 y2 y3 y−12 x3 x5 x7 x1 x8 x2 x4 x6

y2 y−13 y−1

2 y3 n y1 e y−11 x6 x8 x2 x4 x1 x3 x5 x7

y3 y2 y−13 y−1

2 y−11 n y1 e x4 x6 x8 x2 x3 x5 x7 x1

y−12 y3 y2 y−1

3 e y−11 n y1 x2 x4 x6 x8 x5 x7 x1 x3

y−13 y−1

2 y3 y2 y1 e y−11 n x8 x2 x4 x6 x7 x1 x3 x5

x1 x3 x5 x7 x2 x8 x6 x4 e y1 n y−11 y2 y−1

3 y−12 y3

x3 x5 x7 x1 x4 x2 x8 x6 y−11 e y1 n y3 y2 y−1

3 y−12

x5 x7 x1 x3 x6 x4 x2 x8 n y−11 e y1 y−1

2 y3 y2 y−13

x7 x1 x3 x5 x8 x6 x4 x2 y1 n y−11 e y−1

3 y−12 y3 y2

x2 x8 x6 x4 x5 x7 x1 x3 y−12 y−1

3 y2 y3 e y−11 n y1

x4 x2 x8 x6 x7 x1 x3 x5 y3 y−12 y−1

3 y2 y1 e y−11 n

x6 x4 x2 x8 x1 x3 x5 x7 y2 y3 y−12 y−1

3 n y1 e y−11

x8 x6 x4 x2 x3 x5 x7 x1 y−13 y2 y3 y−1

2 y−11 n y1 e

42

Table 6.3: Cayley table of M16(C2 × C4)e y3 n y−1

3 y1 x3 y−11 x6 y2 x5 y−1

2 x2 x1 x′7 x4 x′8y3 n y−1

3 e x3 y−11 x6 y1 x5 y−1

2 x2 y2 x′8 x1 x′7 x4

n y−13 e y3 y−1

1 x6 y1 x3 y−12 x2 y2 x5 x4 x′8 x1 x′7

y−13 e y3 n x6 y1 x3 y−1

1 x2 y2 x5 y−12 x′7 x4 x′8 x1

y1 x3 y−11 x6 n y−1

3 e y3 x4 x′7 x1 x′8 y2 x2 y−12 x5

x3 y−11 x6 y1 y−1

3 e y3 n x′8 x4 x′7 x1 x2 y−12 x5 y2

y−11 x6 y1 x3 e y3 n y−1

3 x1 x′8 x4 x′7 y−12 x5 y2 x2

x6 y1 x3 y−11 y3 n y−1

3 e x′7 x1 x′8 x4 x5 y2 x2 y−12

y2 x5 y−12 x2 x4 x′7 x1 x′8 n y−1

3 e y3 y1 x6 y−11 x3

x5 y−12 x2 y2 x′8 x4 x′7 x1 y−1

3 e y3 n x6 y−11 x3 y1

y−12 x2 y2 x5 x1 x′8 x4 x′7 e y3 n y−1

3 y−11 x3 y1 x6

x2 y2 x5 y−12 x′7 x1 x′8 x4 y3 n y−1

3 e x3 y1 x6 y−11

x1 x′7 x4 x′8 y2 x2 y−12 x5 y1 x6 y−1

1 x3 e y3 n y−13

x′7 x4 x′8 x1 x5 y2 x2 y−12 x3 y1 x6 y−1

1 y−13 e y3 n

x4 x′8 x1 x′7 y−12 x5 y2 x2 y−1

1 x3 y1 x6 n y−13 e y3

x′8 x1 x′7 x4 x2 y−12 x5 y2 x6 y−1

1 x3 y1 y3 n y−13 e

Table 6.4: Cayley table of M16(C2 × C4, Q)e n x1 x3 x2 x4 y5 y−1

5 y1 y−11 y2 y−1

2 y3 y−13 y4 y−1

4n e x3 x1 x4 x2 y−1

5 y5 y−11 y1 y−1

2 y2 y−13 y3 y−1

4 y4x1 x3 e n y5 y−1

5 x2 x4 y−12 y2 y−1

1 y1 y−14 y4 y−1

3 y3x3 x1 n e y−1

5 y5 x4 x2 y2 y−12 y1 y−1

1 y4 y−14 y3 y−1

3

x2 x4 y−15 y5 e n x3 x1 y4 y−1

4 y−13 y3 y−1

2 y2 y1 y−11

x4 x2 y5 y−15 n e x1 x3 y−1

4 y4 y3 y−13 y2 y−1

2 y−11 y1

y5 y−15 x4 x2 x1 x3 n e y3 y−1

3 y−14 y4 y−1

1 y1 y2 y−12

y−15 y5 x2 x4 x3 x1 e n y−1

3 y3 y4 y−14 y1 y−1

1 y−12 y2

y1 y−11 y−1

2 y2 y4 y−14 y−1

3 y3 n e x1 x3 y5 y−15 x4 x2

y−11 y1 y2 y−1

2 y−14 y4 y3 y−1

3 e n x3 x1 y−15 y5 x2 x4

y2 y−12 y−1

1 y1 y−13 y3 y4 y−1

4 x1 x3 n e x2 x4 y−15 y5

y−12 y2 y1 y−1

1 y3 y−13 y−1

4 y4 x3 x1 e n x4 x2 y5 y−15

y3 y−13 y−1

4 y4 y−12 y2 y1 y−1

1 y−15 y5 x2 x4 n e x1 x3

y−13 y3 y4 y−1

4 y2 y−12 y−1

1 y1 y5 y−15 x4 x2 e n x3 x1

y4 y−14 y−1

3 y3 y1 y−11 y−1

2 y2 x4 x2 y5 y−15 x1 x3 n e

y−14 y4 y3 y−1

3 y−11 y1 y2 y−1

2 x2 x4 y−15 y5 x3 x1 e n

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Table 6.5: Cayley table of M16(Q)e n y1 y−1

1 y2 y−12 y3 y−1

3 y4 y−14 y5 y−1

5 y6 y−16 y7 y−1

7n e y−1

1 y1 y−12 y2 y−1

3 y3 y−14 y4 y−1

5 y5 y−16 y6 y−1

7 y7y1 y−1

1 n e y3 y−13 y−1

2 y2 y−17 y7 y−1

6 y6 y5 y−15 y4 y−1

4

y−11 y1 e n y−1

3 y3 y2 y−12 y7 y−1

7 y6 y−16 y−1

5 y5 y−14 y4

y2 y−12 y−1

3 y3 n e y1 y−11 y6 y−1

6 y−17 y7 y−1

4 y4 y5 y−15

y−12 y2 y3 y−1

3 e n y−11 y1 y−1

6 y6 y7 y−17 y4 y−1

4 y−15 y5

y3 y−13 y2 y−1

2 y−11 y1 n e y−1

5 y5 y4 y−14 y−1

7 y7 y6 y−16

y−13 y3 y−1

2 y2 y1 y−11 e n y5 y−1

5 y−14 y4 y7 y−1

7 y−16 y6

y4 y−14 y7 y−1

7 y−16 y6 y5 y−1

5 n e y−13 y3 y2 y−1

2 y−11 y1

y−14 y4 y−1

7 y7 y6 y−16 y−1

5 y5 e n y3 y−13 y−1

2 y2 y1 y−11

y5 y−15 y6 y−1

6 y7 y−17 y−1

4 y4 y3 y−13 n e y−1

1 y1 y−12 y2

y−15 y5 y−1

6 y6 y−17 y7 y4 y−1

4 y−13 y3 e n y1 y−1

1 y2 y−12

y6 y−16 y−1

5 y5 y4 y−14 y7 y−1

7 y−12 y2 y1 y−1

1 n e y−13 y3

y−16 y6 y5 y−1

5 y−14 y4 y−1

7 y7 y2 y−12 y−1

1 y1 e n y3 y−13

y7 y−17 y−1

4 y4 y−15 y5 y−1

6 y6 y1 y−11 y2 y−1

2 y3 y−13 n e

y−17 y7 y4 y−1

4 y5 y−15 y6 y−1

6 y−11 y1 y−1

2 y2 y−13 y3 e n

44

Bibliography

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[2] Bruck, Richard Hubert, A survey of binary systems, Berlin, Springer, (1958)

[3] Srinivasan, Bhama, Ruth Moufang, 1905-1977, The Mathematical intelli-gencer, Volume 6 (June 1984) pp. 51-55

[4] Drapal, Ales, A simplified proof of Moufang’s theorem, Proceddings of theAmerican Mathematical Society, Volume 139, Number 1 (Jan. 2011), pp.93-98

[5] Glauberman, G and Wright, C.R.B. Nilpotence of finite Moufang 2-loops,Journal of Algebra, Volume 8, Issue 4 (April 1968), pp. 415-417

[6] Vojtechovsky, Petr, The smallest Moufang Loop Revisited, Results in Math-ematics 44 (2003), pp. 189-193

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