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

Moufang Loops General theory and visualization of non-associative Moufang loops of order 16

Mikael Stener

Examensarbete i matematik, 15 hp Handledare: Anders Öberg Examinator: Veronica Crispin Quinonez Juni 2016

Department of Mathematics Uppsala University

Moufang Loops General theory and visualization of non-associative Moufang loops of order 16

Thesis by: Mikael Stener Supervisor: Anders Oberg¨ Uppsala 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. A proof of Moufang’s Theorem is presented which implies the important notice of di-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 ones of order 16 and provide a visualization of the multiplication in these loops as has been previously done by Vojtˇechovsk´y[6] with the (only) non-associative Moufang 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 4 2.1 Groupoids, quasigroups and loops ...... 4 2.2 Inverse property and autotopisms ...... 8 2.3 Moufang loops ...... 9

3 Moufang’s Theorem 13

4 Moufang loops of small order 18

5 Visualization of Moufang loops of order 16 27 5.1 M16(D4, 2) ...... 29 5.2 M16(Q, 2)...... 30 5.3 M16(C2 × C4)...... 31 5.4 M16(C2 × C4,Q)...... 33 5.5 M16(Q)...... 34

6 Appendix 35 6.1 Case studies of elements of different order in Moufang loops . . . 35 6.2 Cayley tables for Moufang loops of order 16 ...... 42

1 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 to give the reader this same feeling. A complete answer to the question of what Moufang loops are will be given in the Theory section of this thesis, but a brief answer is that Moufang loops are groups that are not necessarily associative but must 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 Moufang loops in a context with other known algebraic structures1. We notice firstly that the higher up in the chain, the more general the structure. Also, all groups are Moufang loops, which are all loops, which are all quasigroups, which are all groupoids, but the opposite statements are not true. In this thesis, it is the left side of Figure 1.1 that is of interest.

Figure 1.1: Algebraic structures between groupoids and groups Groupoids divisibility associativity

Quasigroups Semigroups identity identity

Loops Monoids Moufang identity

Moufang Loops invertibility associativity

Groups

The name of Moufang loops stems from the german mathematician Ruth

1This image is inspired by the Wikipedia article on Quasigroups

2 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 of Frankfurt where she got her Ph.D. in mathematics at an age of 25. Receiving a lectureship in both Knigsberg and Frankfurt, the logical course of events would have been to become a Privatdozent, a lecturer recognized by the university but receiving no formal salary for it. However, the rise of the Third Reich made it impossible 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 was given a position as Privatdozent in 1946, and a full professorship in 1957, both at the University of Frankfurt. She died in Frankfurt in 1977 [3]. Moufang’s early work was mostly on projective planes and her later years was dedicated to theoretical physics. The focus of this thesis is however on the results of the paper Moufang wrote in 1935: Zur Struktur von Alternativko¨rper, 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 loops by Bruck [2]. In Zur Struktur von Alternativko¨rper, Moufang proves what is called Moufang’s Theorem stating that any three elements a, b, c in a Moufang loop that associate, i.e. where a(bc) = (ab)c, generate a group.

*

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

3 Chapter 2

Theory

The reader of this thesis is assumed to have knowledge of the basic concepts of groups, so comparisons between groups and other structures will be made repeatedly. Regarding notation, a binary operation might be noted with the symbol first defined (usually (·)), but will often instead be written as juxtaposition, for simplicity. Juxtaposition have higher priority than (·). Almost all theory in this chapter is gathered from the textbook Quasigroups and 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 is not the elements themselves that are of most interest, but rather how these elements relate with respect to the binary operation of the structure. A binary operation 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 this operation. We shall now meet the first structure constituting of a set and a binary operation, namely groupoids, the most general structure shown in Figure 1.1. 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 is simply the cardinality |G| of the set G. A groupoid (G, ·) may be noted simply by G, where the binary operation is assumed to be noted by (·) or by juxtaposition. To get a sense of just how general groupoids are, we show in Table 2.1 the Cayley tables (often called multiplication table) of the groupoids of order 2 where all are different up to isomorphism. We notice in particular that only one of the tables is a latin square, i.e. that all elements occur only once in each row and column. In the first groupoid for instance, both 1 · 1 = 1 and 1 · 2 = 1.

4 Table 2.1: Cayley tables of groupoids of order 2 · 1 2 · 1 2 · 1 2 · 1 2 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 2 1 2 2 2 1 2 2 2

· 1 2 · 1 2 · 1 2 · 1 2 1 1 2 1 1 2 1 1 2 1 1 2 2 1 1 2 1 2 2 2 1 2 2 2

This is of course impossible for groups, and we will see soon that this property disappears quickly as we move downwards on the left side of Figure 1.1. An important concept when studying non-associative structures like loops are the translation maps La and Ra. These are defined as

La : G → G, La(x) = a · x and

Ra : G → G, Ra(x) = x · a. Why not simply write left or right multiplication by element a? To understand the reason for this and for future importance, we will study these maps in more 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 as a permutation. The composition of two translation maps LaRb acting on an element 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 the right 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 should now be evident that translation maps are of importance when dealing with non-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 and Ra : 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 have this property. In fact, the only one is the one that is also a group, namely C2. It should also be clear that quasigroups satisfy the cancellation laws; both a · x = a · y and x · a = y · a imply x = y. −1 Definition 2.2 implies that there is an inverse map, noted La such that −1 La(x) = y implies La (y) = x and that the equivalent holds for Ra. These inverse maps are called the conjugates of quasigroups. We use these to define two new binary operations as

5 −1 −1 x\y := Lx (y) and x/y := Ry (x). Due to the bijectivity of the translation maps in quasigroups, the conjugates are unique. It is clear that (G, \) and (G, /) are quasigroups. To connect to the terminology of Figure 1.1, quasigroups can be defined as groupoids where every element is uniquely divisable with another, where divisable in this sense means that a · x = y implies x = a\y and x · a = y implies x = 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 all a, b ∈ G. Definition 2.3 implies that ax = xa for all x and a in G which is the common way to define commutativity, and Definition 2.4 implies that ab · x = a · bx for all a, b, x ∈ G. We now come to the notion of identity elements, which we also define with the help of translation maps. Definition 2.5. Let G be a groupoid and let e ∈ G. Then e is called a left identity element for G if Le is the identity map of G, meaning that Le(x) = x for all x ∈ G. If e is such that Re is the identity map of G, then e is called a right identity element. If e is both a left and right identity element, then we say that e is an identity element.

It will not be proved here, but it should be obvious that if a groupoid has both 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. Let G = {1, 2, 3, 4, 5} and let (·) be given by the following Cayley table.

· 1 2 3 4 5 1 1 2 3 4 5 2 2 1 4 5 3 3 3 5 1 2 4 4 4 3 5 1 2 5 5 4 2 3 1

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

6 element of G. However, since 3 · (3 · 4) = 3 · 2 = 5 6= 4 = 1 · 4 = (3 · 3) · 4, G is not associative, and thus not a group. Speaking of groups, we are now at the bottom of Figure 1.1 and we here define 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 of identity elements to loops and from loops via associativity to groups. We will later come to the core of this thesis, Moufang loops, but we first ask ourselves whether we can go from quasigroups via associativity to some other structure, i.e. are there associative quasigroups that have no identity element and are therefore neither loops nor groups? The following theorem answers this question. Theorem 2.1. If G is a quasigroup which is associative, then G necessarily has an identity element and is thus a group. Proof. We have from the definition of groupoids that G is non-empty, so there is an element a ∈ G. From the definition of quasigroups and the comments following it, there is an element e ∈ G such that a · e = a. Let now b be any element 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, we have Re(b) = be = ya · e = y · ae = ya = b. Since b is any element in G, we thus have 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 so Le is also the identity map on G and by definition, e is an identity element of G. This element is unique since if e0 is also an identity element, then ee0 = e and ee0 = e0 and so e = e0. 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 subloops of 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. This concept is present in groupoids in general and we will define this soon, but we first need another concept very important for non-associative structures, namely the nucleus.

Definition 2.9. The left nucleus Nλ, the middle nucleus Nµ and the right nucles 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)

7 Similar to how the center in non-abelian groups contain the elements that do commute with all other elements, the nucleus of non-associative groupoids contain the elements that do associate with all other pair of elements. We can now 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 commutes and associate with all other elements. We note that the identity element of a group will always be in the center, but a groupoid might not have an identity element, and so the center of a groupoid might be empty. We also see that if a groupoid G is associative, then N(G) = G and if it is also commutative, then Z(G) = G. We can now ask ourselves to which extent elements that are not in the center commute with each other. If neither a nor b are in Z of a groupoid G, then ab 6= ba, but ab = ba · c for some element c. If G is a quasigroup, then c is unique. 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 element d. 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.P λ quasigroup (or L.I.P. loop if it is a loop) if there exists a bijection δλ : x 7→ x such that xλ · xy = y for all x and y in G. A quasigroup/loop is said to be a R.I.P quasigroup/loop (a quasigroup/loop with the right inverse property) if ρ ρ there exists a bijection δρ : x 7→ x such that yx · x = 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 have the inverse property and is called an I.P. quasigroup/loop. An I.P. loop thus satisfies x−1 · xy = y = yx · x−1 where x−1 is, as in groups, called the inverse of x. This property is very important and will play a dominant role when proving Moufang’s Theorem in chapter 3. One of the axioms of groups is that all elements have inverses, so that for all a ∈ G where G is a group there exists an element denoted a−1 such that a · a−1 = e where e is the identity element of the group. This is of course not true for groupoids in general, but for any loop L, we at least have that there for all 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 an element x solves the equation xxρ = e. Any quasigroup can in general have either the right inverse property or the left inverse property. However, the next theorem shows that if a loop has one of them, it has the other.

8 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 has R.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 identity relations of quasigroups. For example, we have that the associative law can be expressed 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 of Moufang loops.

Theorem 2.3. Let σ = (α, β, γ) be an autotopism of an I.P. loop L and let δ −1 be defined as δ(x) = x . 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 an autotopism on L. In the same way, the other statements of the theorem can be proved.

2.3 Moufang loops

We come now finally to the central concept of this thesis, Moufang loops. When Ruth Moufang wrote Zur Struktur von Alternativk¨orper 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 a unique 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))

9 Using the terminology of today, we see that (III) in fact means that Q* has the inverse property. Although Moufang used this property as a definition, we will later use another definition for Moufang loops and then prove that it has the inverse property. Moufang then defined a Quasigroup Q** as a set that satisfies the above conditions, and also satisfies (xy)(zx) = x((yz)x). In the same paper, she showed 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 with another 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 · y xy · x = x · yx yx · 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 associativity since 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 loops was 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 this as well as the equivalence of (M1)-(M5) and the inverse property of Moufang loops in the same theorem. Theorem 2.4. Each Moufang loop M is left and right alternative, flexible and satisfies the inverse property. Also, the Moufang identities are equivalent. Proof. The proof is divided into nine steps (i)-(ix).

10 (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 and due 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- −1 orem 2.3, σλ = (δLxδ, LxRx,Rx), where δ(x) = x , is also an autotopism of a Moufang 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) to the 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 right alternative 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

11 are said to be in the Moufang center of M, denoted C(M). In fact there are commutative non-associative Moufang loops, the smallest of which is of order 81, 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, but have not seen any examples of non-associative Moufang loops. Of course, as stated in the introduction, all groups are Moufang loops, which should be ob- vious considering the definition, but these are all associative. What about the following loop of order 5 we have seen before? Is it Moufang?

· 1 2 3 4 5 1 1 2 3 4 5 2 2 1 4 5 3 3 3 5 1 2 4 4 4 3 5 1 2 5 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 can thus not be a Moufang loop. As we shall see, the smallest Moufang loop is in fact 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 Ruth Moufang in her mentioned paper and is here proved in the next chapter.

12 Chapter 3

Moufang’s Theorem

First a reminder on notation; for any set of elements a1, . . . , an in a loop L we write ha1, . . . , ani as the subloop of L that is generated by a1, . . . an, meaning the subloop that can be constructed only by multiplication and inverses of a1, . . . an. In Zur Struktur von Alternativk¨orper, 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 ha, b, ci 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 elements should 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, the proof follows the outline of AleˇsDr´apal[4] which uses nothing else than the concepts 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

−1 −1 and also that Ra = Ra−1 and La = La−1 , due to

( −1 x = ya = Ra−1 (y) −1 Ra(x) = xa = y ⇐⇒ −1 =⇒ Ra−1 = Ra x = y/a = Ra (y) ( −1 x = a y = La−1 (y) −1 La(x) = ax = y ⇐⇒ −1 =⇒ La−1 = La x = a\y = La (y) 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 assumption leads to α(1)β(x) = γ(1 · x) and thus β(x) = γ(x). We also always have that γ(x) = α(x) · b where b = β(1). Thus, whenever α(1) = 1, we can find an autotopism (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)(α0, β0, γ0) = (αα0, ββ0, γγ0)

13 Lemma 3.1. Let (α, β, γ) be an autotopism of an I.P. loop L. Suppose that α(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). Using α(x) = x and (3.1) we get x · α(x−1)b = b where b = β(1). So α(x−1)b = x−1b and α(x−1) = x−1. Lemma 3.2. Let M be a Moufang loop with an autotopism (α, β, γ) such that α(1) = 1. Then if x, y ∈ M are such that α(x) = x and α(y) = y, then α(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 Moufang loop is an I.P. loop, every element in M can be expressed as multiplication of the elements in X± = {x, x−1; x ∈ X}. Let us now define an element in M as l(x1, . . . , xk) which is equal to 1 if k = 0 and equal to x1l(x2, . . . , xk) if k ≥ 1. For example, 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 if k ≥ 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, we shall 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 obviously true. For n ≥ 2, we put x = u1, s = l(u2, . . . , un) and t = l(v1, . . . , vm). The induction assumption now is that s · t = l(u2, . . . , un, v1, . . . , vm) and we will prove that if this is true, then xs · t = l(u1, . . . , un, v1, . . . , vm). We express xs · t as xs · (tx−1 · x) = x(s · tx−1)x due to Moufang identity (M2) and flexibility. −1 −1 −1 −1 Now, tx = r(v1, . . . , vm)x = r(v1, . . . , vm, x ) = l(v1, . . . , vm, x ) and so −1 xs · t = x(s · l(v1, . . . , vm, x ))x. And so, by the induction assumption, we have −1 xs · t = xl(u2, . . . , un, v1, . . . , vm, x )x −1 = x(r(u2, . . . , un, v1, . . . , vm)x )x = xl(u2, . . . , un, v1, . . . , vm)

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

14 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, generated by X±. S is also a group since all generating elements have inverses. Thus S = M.

−1 Lemma 3.4. Let x and y be elements of a Moufang loop M. Then Lxy LxLy = −1 −1 −1 [Rx ,Ly] and Rxy RxRy = [Lx ,Ry]. −1 −1 −1 Proof. The first identity states that Rx Ly = LyRx · Lxy LxLy. Cancelling Ly and rewriting gives LxRxLy = LxyRx which, acting on an element z ∈ M gives x((yz)x) = (xy)(zx), which is Moufang identity (M2), so the first equality is true −1 in any Moufang loop. As for the other identity to be proved, we have Lx Ry = −1 −1 RyLx · Rxy RxRy ⇐⇒ RyxLx = RxLxRy ⇐⇒ (xz)(yx) = (x(zy))x and we have Moufang identity (M5).

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

±1 Lemma 3.5. Let M be a Moufang loop. Suppose that α = [Rx,Ly] where x, y ∈ M or that α = Lε1 ...Lεn where x , . . . , x ∈ M and ε , . . . , ε ∈ −1, 1. x1 xn 1 n 1 n Then there exist β and γ such that (α, β, γ) is an autotopism of M.

−1 −1 Proof. We have that Rx = Rx−1 and Lx = Lx−1 so together with Lemma 3.4 ±1 we have an autotopism if α = [Rx,Ly] . To prove the statement for the case α = Lε1 ...Lεn we use that the Moufang identity xy · zx = x(yz · x) states that x1 xn (Lx,Rx.LxRx) is an autotopism for every x ∈ M. So we have an autotopism −1 when α = Lx and then of course when α = Lx . An autotopism for when α = Lε1 ...Lεn for arbitrary n can thus be obtained as the composition of x1 xn εi autotopisms (Lxi ,Rxi ,Lxi Rxi ) where 1 ≤ i ≤ n.

Lemma 3.6. Let x1, x2, x3 be elements of a Moufang loop M. If x1 · x2x3 = x1x2 · x3, then ε1 ε2 ε3 ε1 ε2 ε3 xσ(1) · xσ(2)xσ(3) = xσ(1)xσ(2) · xσ(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 and left and right alternative which already has been proven. Let us now assume that M is generated by x = x1, y = x2 and z = x3 and that x · yz = xy · z. −1 −1 By Lemmas 3.4 and 3.5, any of the mappings Lxy LxLy, Ryz 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 shown that, given any permutation σ ∈ S3 and some ε1, ε2 and ε3, the equality in the

15 0 lemma hold for the same σ and for any other εi ∈ {−1, 1}. We now need to 0 show that it also holds for all other permutations σ ∈ S3 given σ. We do this by showing that it holds for any two permutations that generate S3, namely −1 −1 a 3-cycle and a transposition. The identity Lxy LxLy = [Rx ,Ly] in Lemma 3.4 means that x · yz = xy · z implies y · zx−1 = yz · x−1 and due to the first part of this lemma, this implies y · zx = yz · x, so we have a 3-cycle. For our transposition, we use the fact that each Moufang loop has the inverse property and thus (x · yz)−1 = (xy · z)−1 ⇐⇒ z−1y−1 · x−1 = z−1 · y−1x−1 which, again by the first part of this lemma, implies zy · x = z · yx. So the equality that is to be proven holds for permutations σ = (1 2 3) and τ = (1 3) who together generate 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 equalities contain 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 Lemma 3.6 and the induction axiom. We can thus assume that, say, u0 = x and uk+1 = y. We will now prove the equality for the different cases of u1 and uk, ±1 namely u1, uk ∈ {x, y, z} . We put s = l(u2, . . . , uk). The equality is then equivalent to x(u1s · y) = −1 −1 −1 −1 (x·u1s)y. If u1 = x , then x(x s·y) = (x·x s)y ⇐⇒ x(x s·y) = sy ⇐⇒ x−1s · y = x−1 · sy and the last equality holds by the induction assumption. If −1 −1 −1 instead u0 = x and u1 = x, we have x (xs · y) = (x · xs)y, and due to Lemma 3.6 we have x(xs · y) = (x · xs)y, viewing xs as one element. We ±1 have shown that the equality holds for u1 = x . By the induction assumption, l(u1, . . . , uk) = r(u1, . . . , uk) so the situation is left-right symmetric. The case ±1 uk = y can thus be shown by mirror reasoning. By the induction assumption, xs · y = x · sy, and by Lemma 3.6, this implies xy · s = x · ys. Right-multiplying by y, we get (x · ys)y = (xy · s)y = x(ys · y) by a Moufang identity. Again by Lemma 3.6 we have x · sy)y−1 = x(ys · y−1, and −1 −1 ±1 so we have also (x · y s)y = x(y s · y). This solves the case u1 = y and the ±1 mirror argument solves the case uk = x . ±1 ±1 We have left to show the identity for u1 = z and uk = z . It will suffice −1 to consider the case u1 = uk = z and the case u1 = z and uk = z . We put w = 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, we have 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. that −1 −1 −1 −1 α(z ) = z where α = Lw Lz [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 have α(z) = z from x(zwz) · y = x · (zwz)y. Then by Lemmas 3.1 and 3.5 we have proven 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 first equality 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. If x · 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 elements will also associate, thus a group is generated. Corollary 3.1. Any Moufang loop M is di-associative, meaning that any two elements 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 any element 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 the remainder of this thesis, and especially in Chapter 5, Corollary 3.1 will have the most impact in the Moufang loops we study.

17 Chapter 4

Moufang loops of small order

Since all groups are Moufang loops, the right term of the subject of this thesis is non-associative Moufang loops. However, for simplicity, we will sometimes use Moufang loops even when we mean non-associative Moufang loops, and the meaning will be evident from the context. This Chapter follows the outline of Chein [8] and additional clarifications have been added when necessary. As stated in the introduction, the purpose of this thesis is to get a general understanding of Moufang loops. A good way to do this is to study Moufang loops of small order. It was earlier stated that the smallest Moufang loop is of order 12. Table 4.1 [7] shows the number of non-associative Moufang loops of order ≤ 32 and so a good limitation in this thesis is to study the Moufang loops of order ≤ 31.

Table 4.1: Number π(n) of non-associative Moufang loops of order n Order n π(n) 12 1 16 5 20 1 24 5 28 1 32 60

In order to study Moufang loops of order ≤ 31 then it will be obvious that it is crucial with an understanding of non-cyclic groups of order ≤ 15. We show in Table 4.2 these groups together with the number of elements in the minimal set 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 and x∈ / H, let d be the smallest divisor larger than 1 of |x| such that |x|/d divides H. Then |hH, xi| ≥ d · |H|. Proof. We consider all elements of the form hxi where h ∈ H and 0 ≤ i < d. If i j j −i j−i j−i −1 h1x = h2x for i ≤ j we have h1 = (h2x )x = h2(x ), so x = h2 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 that t 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. So since d is chosen as the smallest divisor > 1 of |x| such that |x|/d divides H we either 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 and i j i thus whenever h1x = h2x we have i = j and h1 = h2. So the elements hx , h ∈ H, 0 ≤ i < d are distinct and there are |H| · d of them. We have of course hxi ∈ hH, xi, but these elements are not necessarily the only ones in hH, xi and so the lemma is proven. This lemma might seem difficult to grasp, but Corollary 4.3 provides a good example on how it can be used in practice.

Corollary 4.1. If |hH, xi| = d · |H|, then every element of hH, xi may be uniquely expressed in the form hxi, h ∈ H, 0 ≤ i < d. Proof. Since the elements hxi are distinct and since there are exactly d · |H| = |hH, xi| of them, they are all the elements in hH, xi. Corollary 4.2. If H and x are as in Lemma 4.1 and |x| is prime, then |hH, xi| ≥ |x||H|. Proof. This should be obvious, since d = |x|. Corollary 4.3. If a nonassociative Moufang loop M of order ≤ 31 contains an element z of order > 3, and if x, y ∈ M, y∈ / hzi and x∈ / hy, zi, then M = hx, y, zi.

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

|hx, y, zi| ≥ 2|hy, zi| ≥ 4|z| ≥ 16.

If M 6= hx, y, zi, then there exists w ∈ M such that w∈ / hx, y, zi. But then |M| ≥ |hw, x, y, zi| ≥ 2 · |hx, y, zi| ≥ 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 Moufang loops cannot be generated by two elements. So we now know that we are looking for loops generated by three elements or, if the loop is to have a minimal generating 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 technicalities we have not gone through in this thesis. We state it here mainly because of one of 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 3 2 Proof. If |M| = p then |M × Cp2 | = p and if |M| = p then |M × Cp| = p3. Either way, M is isomorphic to a subloop of the direct product which, by Proposition 4.1 is a group. Any subloop of a group is associative, so M is a group. In the search for Moufang loops M of order ≤ 31 we will consider all cases of the possible order of the elements in minimal sets of generators of M. We start with the important case in which every set of generators of M contains an element of order 2. Theorem 4.1. If M is a nonassociative Moufang loop for which every minimal set of generators contains an element of order 2, then there exists a nonabelian group G, and an element x of order 2 in M, such that each element of M may be uniquely expressed in the form gxα, where g ∈ G, α ∈ {0, 1}, and the product of two elements of M is given by

δ ε ν µ ν δ+ε (g1x )(g2x ) = (g1 g2 ) x (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 the fewest possible elements of order 2. We can assume that x is of order 2. We let G = hu1, . . . , uni. If g ∈ G then {gx, u1, . . . , un} generate M. Now, by the way 2 x, u1, . . . , un were chosen, gx must be of order 2. Hence, (gx) = gxgx = 1, so

gx = x−1g−1 = xg−1 (4.2)

ε For the sake of wanting to multiply any elements of the form gix , we will ε first study the multiplication g1(x g2), ε ∈ {0, 1}. We have, due to the inverse property and a Moufang identity:

ε −1 ε −1 ε g1(x g2) = (g2(g2 g1))(x g2) = (g2((g2 g1)x ))g2

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

−1 −1 −1 −1 −1 −1 g1(xg2) = (g2((g2 g1)x))g2 = (g2((g2 g1)x)) g2 = (((g2 g1)x) g2 )g2 −1 −1 −1 −1 −1 −1 −1 −1 = ((g2 g1)x) = x (g2 g1) = (g2 g1)x = (g1 g2) x So we see that regardless of the value of ε, we have

ε ν ν ε g1(x g2) = (g1 g2) x (4.3) where ν = (−1)ε.

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

δ ε δ ε δ −δ δ ε+δ −δ (g1x )(g2x ) = (((g1x )(g2x )x )x = (g1(x g2x ))x

ε+δ δ ε+δ δ −ε−δ −1 Now if ε and δ are such that x = x then, due to (4.2), x g2x = x x g2 = ε −1 ε+δ 0 2 δ ε+δ δ ε x g2 . If however, x = x = x = 1, then ε = δ and x g2x = x g2 = x g2. So we have

δ ε ε µ −δ (g1x )(g2x ) = (g1(x g2 ))x where µ = (−1)ε+δ. Hence, by (4.3),

δ ε ν µ ν ε −δ ν µ ν ε+δ (g1x )(g2x ) = ((g1 g2 ) x )x = (g1 g2 ) x We can conclude that the product of two elements of the form gxα, g ∈ G, is again of that form. So the elements of the form gxα form a subloop of M. But since all elements x, u1, . . . , un can be expressed in this form, this subloop is all of M. α β Every element has a unique representation in this form, since if g1x = g2x α−β −1 where α, β ∈ {0, 1}, then x = g1 g2 ∈ G. However, u1 . . . , un cannot generate M, so x∈ / G and thus α − β = 0 and α = β. Therefore, g1 = g2 and the expression is unique. We now need to show that elements of the form gxα, under the operation defined by (4.1), form a nonassociative Moufang loop if and only if G is a δ ε nonabelian group. We first clarify the product of two elements (g1x )(g2x ) for the three different cases when (δ, ε) 6= (0, 0).

−1 (g1x)(g2x) = g2 g1 −1 (g1x)g2 = (g1g2 )x g1(g2x) = (g2g1)x

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

−1 −1 (ab)(ca) = (g1g2)((g3x)g1) = (g1g2)((g3g1 )x) = ((g3g1 )(g1g2))x And also

(a(bc))a = (g1(g2(g3x)))g1 = (g1((g3g2)x))g1 −1 = (((g3g2)g1)x)g1 = (((g3g2)g1)g1 )x = (g3g2)x

−1 −1 −1 −1 So (g3g1 )(g1g2) = (g3g2) and ((g3g1 )(g1g2))g1 = (g3g2)g1 . Thus, due to a Moufang identity and the inverse property

−1 −1 −1 −1 g3(g1 (g1g2)g1 ) = g3(g2g1 ) = (g3g2)g1

22 for any g1, g2, g3 ∈ G. Hence, when M is Moufang, G is an associative subloop of α β M and therefore a group. To see when M is associative, let a = g1x , b = g2x γ and c = g3x . Then

ν1 µ1 ν1 α+β γ ν1 µ1 ν1ν2 µ2 ν2 α+β+γ (ab)c = ((g1 g2 ) x )(g3x ) = ((g1 g2 ) g3 ) x ,

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

α ν3 µ3 ν3 β+γ ν4 ν3 µ3 ν3µ4 ν4 α+β+γ a(bc) = (g1x )((g2 g3 ) x ) = (g1 (g2 g3 ) ) x ,

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

ν1 µ1 ν1ν2 µ2 ν2 µ3 ν2 µ3 ν2µ2 µ3 ((g1 g2 ) g3 ) = (g1 (g2 g3 ) )

−1 −1 −1 −1 −1 Taking α = β = 0, γ = 1, g3 = 1 we get ((g1g2) ) = (g1 (g2 )) , which becomes g1g2 = g2g1. So if M is associative then G is abelian. Conversely, if G is abelian, then

ν1 µ1 ν1ν2 µ2 ν1 ν1 µ1 ν1 µ2ν2 µ1ν1 µ2ν2 ((g1 g2 ) g3 ) = (g1 g2 ) g3 = g1g2 g3 , using that ν1ν1 = 1. We also have

µ3 ν2 µ3 ν2µ2 µ3 µ2µ3 ν2µ2 (g1 (g2 g3 ) ) = g1g2 g3 , α using that µ3µ3 = ν2ν2 = 1. We have that µ1ν1 = (−1) = 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. α β γ We must check the Moufang identity. We put a = g1x , b = g2x , c = g3x . Then

ν1 µ1 ν1 α+β ν2 µ2 ν2 α+γ ν1 µ1 ν1ν3 ν2 µ2 ν2µ3 ν3 β+γ (ab)(ca) = ((g1 g2 ) x )((g3 g1 ) x ) = ((g1 g2 ) (g3 g1 ) ) x and

α ν4 µ4 ν4 β+γ α (a(bc))a = ((g1x )((g2 g3 ) x ))(g1x )

ν5 ν4 µ4 ν4µ5 ν5 α+β+γ α = ((g1 (g2 g3 ) ) x )(g1x )

ν5 ν4 µ4 ν4µ5 ν5ν6 µ6 ν6 β+γ = ((g1 (g2 g3 ) ) g1 ) x where

α+β β µ1 = (−1) , ν1 = (−1) α+γ α µ2 = (−1) , ν2 = (−1) β+γ α+γ µ3 = (−1) , ν3 = (−1) = µ2 β+γ γ µ4 = (−1) = µ3, ν4 = (−1) α+β+γ β+γ µ5 = (−1) , ν5 = (−1) = µ3 β+γ α µ6 = (−1) = µ3, ν6 = (−1) = ν2

23 So we need

ν1 µ1 ν1ν3 ν2 µ2 ν2µ3 ν3 ν5 ν4 µ4 ν4µ5 ν5ν6 µ6 ν6 ((g1 g2 ) (g3 g1 ) ) = ((g1 (g2 g3 ) ) g1 ) for all cases of the values of α, β and γ. These eight cases are shown in Table 4.3.

Table 4.3: Test of Moufang identity in M2n(G, 2) α β γ (ab)(ca) (a(bc))a

−1 −1 −1 −1 −1 −1 −1 −1 −1 0 0 1 ((g1g2) (g3g1 ) ) = g3g2 (g1 (g2 g3 )) g1 = g3g2 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 0 1 0 (g1 g2 ) (g3g1) = g2g3 (g1 (g2g3 ) ) g1 = g2g3 −1 −1 −1 −1 −1 −1 −1 −1 0 1 1 ((g1 g2 )(g3g1 )) = g1g3 g2g1 (g1(g2 g3) )g1 = g1g3 g2g1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 1 0 0 ((g1g2 ) (g3 g1 ) ) = g3 g2 ((g1(g2g3) ) g1) = g3 g2 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 1 0 1 ((g1g2 )(g3 g1)) = g1g2 g3 g1 ((g1 (g2 g3 ) )g1 ) = g1g2 g3 g1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 1 1 0 ((g1 g2)(g3 g1 )) = g1g3g2 g1 (g1 (g2g3 )g1 ) = g1g3g2 g1 −1 −1 −1 −1 −1 −1 −1 −1 −1 1 1 1 (g1 g2) (g3 g1) = g2 g3 ((g1(g2 g3)) g1) = g2 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 set of a Moufang loop contains an element of order 2. It also gives rise to a method for finding Moufang loops of order 2n, where n is the order of a non-abelian group G. This class of Moufang loops is very important and a loop constructed in 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 which every 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 minimal sets of generators contain element of order 2. So, for every yz ∈ M, there exists an x∈ / hy, zi such that |x| > 2. Proposition 4.2. If a nonassociative Moufang loop M contains an element x such that |x| ≥ 8, then |M| > 31. Proof. There must exist y, z ∈ M such that y∈ / hxi and z∈ / hx, yi. By Lemma 4.1, |M| ≥ |hx, y, zi| ≥ 2|hx, yi| ≥ 4|hxi| ≥ 32.

24 Proposition 4.3. The only nonassociative Moufang loops M of order ≤ 31 which 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 order 2, then the result follows from Corollary 4.5. Assume now that there exists a minimal set S of generators each of which is of order greater than 2. Let a, b ∈ S. We note that ha, bi is not cyclic, since then S would not be minimal. Since S is not associative, M 6= ha, bi, and so, by Lemma 4.1, ha, bi ≤ 15. If z ∈ ha, bi, a survey of Table 4.2 implies that ha, bi = D5 or D7. But then either a or b would be of order 2, contrary to assumption. So z∈ / ha, bi. By Corollary 4.2, M = hz, a, bi and ha, bi ≤ 6. But then either a or b is of order 2, again contrary to assumption. We can thus conclude that the assumption that S does not contain an element of order 2 is false and the proposition follows. The goal now is to find all non-associative Moufang loops of order less than 32 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 elements in 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 however very technical and not very enjoyable, so the calculations have been put in the Appendix, but the results of these cases are here presented. Case 1 gives rise to the loops M24(G12,C2×C4) and M24(G12,Q) represented by

2 2 3 6 M24(G12,C2 × C4) =hx, y, z : x = y = z , z = 1, ((xαyβ)zγ )((xσyτ )zρ) = (xα+σyβ+τ )zµγ+νρ, where µ = (−1)σ+τ+στ and ν = (−1)ατ+βσi.

2 2 3 6 M24(G12,Q) =hx, y, z : x = y = z , z = 1, ((xαyβ)zγ )((xσyτ )zρ) = (xα+σyβ+τ )z3βσ+µγ+νρ, where µ = (−1)σ+τ+στ and ν = (−1)ατ+βσi.

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

25 2 2 2 4 M16(C2 × C4) =hx, y, z : x = y = z ; z = 1; (xαyβ)zγ · (xσyτ )zρ = (xα+σyβ+τ )zµγ+νρ, where µ = (−1)στ and ν = (−1)ατ+βσi

2 2 2 4 M16(C2 × C4,Q) =hx, y, z : x = y = z ; z = 1; ((xαyβ)zγ )((xσyτ )zρ) = (xα+σyβ+τ )zµγ+νρ+2βσ, where µ = (−1)στ and ν = (−1)ατ+βσ

2 2 2 4 M16(Q) =hx, y, z : x = y = z ; z = 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 Moufang loops of order 16

Many groups of small order have fairly simple multiplication rules, making it easy to get a sense of the structure of them. But the non-associative Moufang loops 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 Vojtˇechovsk´y[6] presents graphs for the multiplication in M12(S3, 2) in a way that visualize the multiplication between all of the elements in the loop. The graphs also illustrate the di-associativity of Moufang loops by showing what subgroups all pairs of elements generate. These graphs do not follow any common standard and will here be referred to as multiplication graphs. Table 5.1 [8] shows non-associative Moufang loops of order 12 and 16 along with the subgroups generated by two or more elements, nucleus N and number of elements of order 2,3 and 4. All loops have of course subgroups generated by one element, but how these are generated is obvious. As we present multiplication graphs 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 Order of Subgroups generated by elements Moufang loop two or more elements 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 Appendix with the name of the elements corresponding to how we name them in this chapter. We thus leave the notation of the elements in the loops used in Chapter 4. Except for in the case of M12(S3, 2), the graphs of the loops in this chapter are 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. The involutions are named x1, . . . x9 and the elements of order 3 are named y and y−1. The involutions are then put as vertices in a circle as shown in Figure 5.1. Multiplication between two involutions are illustrated by an edge and if that edge is solid, then the product of those elements is the third vertex in the triangle that the multiplication gives rise to. If two involutions are connected by a dotted line, then multiplication between these elements result in an element of order 3. This multiplication we call clockwise positive, meaning that, for −1 example x1 · x4 = y and x4 · x1 = y .

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

x1 x1 x1 x1 x9 x2 x9 x2 x9 x2 x9 x2

x8 x3 x8 x3 x8 x3 x8 x3 y±1

x7 x4 x7 x4 x7 x4 x7 x4

x6 x5 x6 x5 x6 x5 x6 x5

The subgroup structure of M12(S3, 2) is now evident. Any two elements xi, 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 multiplication between xi and xj gives rise to, as well as the identity element e, not shown in the graphs. Any two elements xi, xj connected by a dotted edge generate S3. This subgroup then consists of xi, xj, the third involution in the dotted triangle that −1 multiplication between xi and xj gives rise to, as well as e, y and y . We now use this way of constructing multiplication graphs on the Moufang loops 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 so the involutions of M16(D4, 2) are {x1, . . . , x12, n}. We treat multiplication by n differently, so the vertices in the multiplication graph of M16(D4, 2) are only x1, . . . , x12. The elements are then ordered in such a way that the product of involutions on the opposite side of each other is n. Graphs I-III in Figure 5.2 shows multiplication between all involutions x1, . . . , x12. As before, a solid edge between two elements implies that the product of these elements is the element in 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 x1 x1 x1 I x12 x2 II x12 x2 III x12 x2 IV x12 x2

x11 x3 x11 x3 x11 x3 x11 x3

±1 x10 x4 x10 x4 x10 x4 x10 y x4

x9 x5 x9 x5 x9 x5 x9 x5

x8 x6 x8 x6 x8 x6 x8 x6 x7 x7 x7 x7 All pairs of elements that are connected by a solid edge in graphs I-III in Figure 5.2 generate V4. The elements of V4 is then the elements in a triangle and 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 in Figure 5.2, we see the multiplication of involutions such that the product is an element of order 4. in fact, the multiplication is clockwise positive, so x1 ·x4 = y −1 and x4 · x1 = y as before. We thus see that for any xi, xj connected by a ∼ dotted edge, hxi, xji = D4 and the elements in D4 are then all elements in the −1 square where xi and xj are vertices, as well as y, y , 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. How are these generated? We also see that it has nine involutions and six elements of order 4. However, as in M16(D4, 2), we have two elements in the nucleus, e and the one we call n which again is an involution. The elements of order 4 are −1 −1 −1 denoted y1, y1 , y2, y2 , y3, y3 .

Figure 5.3: Multiplication and generating sets in M16(Q, 2) y I x1 II x1 III x1 IV 1 x x x x x x 8 2 8 2 8 2 −1 y y3 2 x ±1 x x ±1 x x ±1 x 7 y1 3 7 y2 3 7 y3 3 −1 y2 y3 x6 x4 x6 x4 x6 x4 −1 x5 x5 x5 y1

The multiplication graphs I-III in Figure 5.3 have involutions x1, . . . , x8 as vertices. We see that if two involutions are connected by a directed edge, then the product of these elements is one of the elements of order 4 in the center of the graph. Multiplication in the direction of the arrows is ”positive”, so that, −1 for example, x1 ·x2 = y2 and x2 ·x1 = y2 . As before, the product of involutions on the opposite side of each other is n. Two involutions that are connected by a directed edge generate all four involutions in the rectangle wherein they are vertices, as well as the elements of order 4 in the center of that rectangle. They also generate e and n and so ∼ hxi, xji = D4 where xi and xj are connected by a directed edge. Two involutions on the opposite side of each other generate V4. In the previous loops, we have only two elements of order 3 or 4. Now that we have six elements of order 4, we show the multiplication of these elements in graph IV of Figure 5.3. The graph shows six directed 3-cycles. Multiplication of two elements in the direction of the 3-cycle results in the third element of that cycle. Note that all pairs of elements that are not on the opposite side of each other are contained in two different 3-cycles, and that, for example y1 · y2 = y3 −1 but y2 · y1 = y3 . We have that any two elements in graph IV in Figure 5.3 that are not inverses generate all the other elements in the graph, as well as e and n (since ±1 2 ±1 ±1 ∼ (yi ) ∈ {e, n} for any i). Thus, hyi , yj i = Q for any yi, yj such that ±1 ±1 ±1 ±1 yi 6= yj or yi · yj 6= e.

30 5.3 M16(C2 × C4)

Figure 5.4: The multiplication graphs we use to describe the mul- tiplication in Moufang loops in this thesis do not fol- x1 low an exact standard. The graphs we have seen so

x6 x2 far could have looked different if the vertices was cho- sen to represent other elements in the loop or if they were put in a different order. So far, it has however been a quite obvious choice of how to represent the x5 x3 loops. Turning now to the loop M16(C2 × C4) we will x4 see that the choice of vertices are a bit different but the underlying reason for this is to produce graphs that are most easily read and that show the symmetry of the loops in the clearest way. We will now abandon the objective of trying to find multiplication rules for all of the elements in M16(C2 × C4), due to the more complicated structure of this 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 do have a simple multiplication rule similar to the ones of involutions in the previous loops. The product of two elements joined by an edge is the third element in ∼ that triangle. And as before, hxi, xji = V4 if xi and xj are connected by an edge.

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

x1 x1 x1 I x6 x2 II x6 x2 III x6 x2 −1 y −1 y −1 y y2 1 y2 1 y2 1

0 0 0 0 0 ±1 0 x8 x7 x8 x7 x8 y3 x7 −1 y −1 y −1 y y1 2 y1 2 y1 2 x1 x1 x5 x3 x6 x2 x5 x3 x6 x2 x5 x3 x4 x4 x4 −1 y y y y2 1 4 1

0 0 0 ±1 0 IV x8 x7 V x8 y3 x7 −1 y y y y1 2 3 2 x5 x3 x5 x3 x4 x4

Looking now at Figure 5.5, x1, . . . , x6 are kept as vertices in the graphs, −1 −1 and four elements of order 4 are added; y1, y1 , y2 and y2 , as well as two 0 0 involutions, denoted x7 and x8. In graphs I and II, any two elements joined by an edge generate D4 where the elements are all of the elements that are vertices joined by an edge, as well as e and n, where n again is the involution in the nucleus. In graph III in Figure 5.5, we have that the product of two elements joined −1 by an edge is one of the elements in the middle. These elements y3 and y3 are the 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 that are vertices joined by an edge, as well as e and n. In graph V, the clockwise product of elements joined by a dotted edge is y3, while the counter-clockwise −1 product is y3 . 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 that generate which subgroup. In the graphs, the vertices was chosen and ordered so that the subgroup structure was most evident. The twelve vertices are the four ±1 ±1 ±1 ±1 involutions x1, . . . , x4 and the eight elements of order 4, y1 , y2 , y3 , y4 .

Figure 5.6: Elements in M16(C2 × C4,Q) generating C2 × C4 x x x x −1 1 y −1 1 y −1 1 y −1 1 y I y4 1 II y4 1 III y4 1 IV y4 1 −1 y −1 y −1 y −1 y y3 2 y3 2 y3 2 y3 2

x4 x2 x4 x2 x4 x2 x4 x2

−1 y −1 y −1 y −1 y y2 3 y2 3 y2 3 y2 3 −1 −1 −1 −1 y y4 y y4 y y4 y y4 1 x3 1 x3 1 x3 1 x3

Figure 5.7: Elements in M16(C2 × C4,Q) generating Q or D4 x x −1 1 y −1 1 y I y4 1 II y4 1 −1 y −1 y y3 2 y3 2 x ±1 x x ±1 x 4 y5 2 4 y5 2 −1 y −1 y y2 3 y2 3 −1 −1 y y4 y y4 1 x3 1 x3

Figure 5.6 shows which elements that generate C2 ×C4. Any pair of elements that are connected by an edge generate all other elements connected by the edges in that figure, as well as e and n. In Figure 5.7 we have the same multiplication rule as seen before. Clockwise multiplication of two elements connected by a dotted or dashed edge results in y5, another element of order 4, while the −1 counter-clockwise product is y5 . Any pair of elements connected by a dotted line in graph I in Figure 5.7 generate the group Q, while any pair of elements connected by a dashed line in graph II generate D4. It is not shown graphically, but any involutions xi, xj such that xi · xj = n generate V4.

33 5.5 M16(Q)

1 We have reached the last visualization of Moufang loops in this thesis, M16(Q) . Table 5.1 shows that the only subgroup of M16(Q) generated by two elements is −1 Q. We thus have that any two elements a, b ∈ M16(Q) such that a 6= b, a 6= b or a, b∈ / N generate Q. Figure 5.8 shows this fact graphically. All elements not ±1 ±1 in the nucleus is of order 4 so we denote them by y1 , . . . , y7 . The first graph −1 shows that, for example, y1 · y4 = y7 and y4 · y1 = y7 in the same way as seen previously.

Figure 5.8: Elements in M16(Q) generating Q y y −1 1 y −1 1 y y6 2 y6 2 −1 y −1 y y5 3 y5 3 −1 ±1 y −1 y y4 y7 4 y4 4 −1 y −1 y y3 5 y3 5 y1 −1 y1 −1 y1 −1 y y y6 −1 y y y6 −1 y y6 2 2 −1 y6 2 2 −1 y6 2 y1 y1 −1 y −1 y −1 y y5 3 y5 3 y5 3 −1 y −1 y −1 y y4 4 y4 4 y4 4 −1 y −1 y −1 y y3 5 y3 5 y3 5 −1 y −1 y −1 y y2 −1 6 y2 −1 6 y2 −1 6 y1 y1 y1

1 The non-associative Moufang loop M16(Q) is commonly known as the standard basis of the 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∈ / hzi, then, since M cannot be generated by only two elements, M contains an element y∈ / hx, zi. By Lemma 4.1,

31 ≥ |M| ≥ |hx, y, zi| ≥ 2|hx, zi|, so |hx, zi| ≤ 15. Since |z| = 6 we have that hx, zi is a group containing an ∼ element of order 6. Studying Table 4.2, we see that we have hx, zi = 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∈ / hzi ∼ ∼ ∼ and x∈ / hy, zi. In this case, hx, yi = hx, zi = hy, zi = C2 × C6. A knowledge of 2 C2 × C6 tells us that we can, independently of z, choose x and y such that x = 2 2 ∼ y = z , so we can assume that this equality holds. Now since hx, yi = C2 × C6, 2 2 ∼ and x = y we have |xy| = 6. So hxy, zi = 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 Moufang loop. Case 1(b). There exists y∈ / hzi such that |y| = 6, but for every x∈ / hy, zi, ∼ |x|= 6 6. Since hx, zi = C2 × C6,D6 or G12, |x| = 2 or 4. (A knowledge of the groups in question tells us that |x|= 6 3). Now if |x| = 2 for each x∈ / hy, zi, then

35 every minimal set of generators of M contains an element of order 2, resulting in a Moufang loop we have already found. Hence |x| = 4 for some x∈ / hy, zi. ∼ 2 2 hy, zi = C2 × C6, and we can again assume that y = z . We m¡ust have ∼ ∼ 2 3 2 3 3 3 that hx, zi = hx, yi = G12, and thus x = z and x = y implying y = z . But then y = z, contrary to assumption, so case 1(b) cannot occur. Case 1(c). For each w∈ / hzi, |w| < 6. Again, we have that |w| = 2 or 4, and since 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 = hx, y, zi. ∼ ∼ ∼ From Table 4.2 we see that hx, zi = hy, zi = G12 and hx, yi = C2 × C4, Q, or ∼ G12. However, if hx, yi = G12, then there exists an element of order 6 which is ∼ not in hzi, contrary to assumption. So hx, yi = C2 × C4 or Q. In either case, x2 = y2 = z3 and zx = xz−1 and zy = yz−1. ∼ 2 ∼ If hx, yi = C2 × C4, then (xy) = 1, so hxy, zi = C2 × C6 or D6. In the former case |(xy)z| = 6, contradicting that all w∈ / hzi have order less than 6, ∼ −1 so we must have hxy, zi = D6 and thus z(xy) = (xy)z . ∼ ∼ If instead hx, yi = Q, then |xy| = 4 and so hxy, zi = 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, for example,

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

Since x, y and z are on this form and M = hx, y, zi then if we can show that the product of two elements of this form is again of this form, every element of M is expressible in this form. We have that |M| = |hx, y, zi| ≥ 2|hy, zi| = 24 and there are at most 24 distinct element of this form, so every element is then uniquely expressible in this form. To investigate the multiplication of (xαyβ)zγ by (xσyτ )zρ, let u = xαyβ and v = xσyτ . If v 6= 1 (i.e. v = x, y or xy), then zv = vz−1 and thus ( uz2γ+ρ if v = 1 ((uzγ )(vzρ))zγ = u(zγ vzγ+ρ) = . 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 then u2 = 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

 γ+ρ uz if v = 1  ρ−γ γ ρ vz if v 6= 1, u = 1 (uz )(vz ) = 2 ρ−γ u z if u = v 6= 1  (uv)z−ρ−γ otherwise

Now setting ( ( 1 if v = 1 1 if u = 1, or v = 1, or u = v µ = and ν = −1 otherwise −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 ( 1 if σ = τ = 0 µ = = (−1)σ+τ+στ −1 otherwise and ( 1 if α = β = 0, or σ = τ = 0, or α = σ, β = τ ν = = (−1)ατ+βσ −1 otherwise

∼ ∼ We need now to consider the two different cases hx, yi = C2 ×C4 and hx, yi = Q. ∼ Case 1(c1). If hx, yi = 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 achieved Moufang loop has the following representation:

M = hx, y, z : x2 = y2 = z3, z6 = 1, ((xαyβ)zγ )((xσyτ )zρ) = (xα+σyβ+τ )zµγ+νρ, where µ and ν are defined as abovei.

It is now needed to check that this loop is in fact Moufang, but this is, as seen before, very tedious and will be omitted. We can however easily see that M 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). ∼ 3 2 Case 1(c2). We have here that hx, yi = Q, so that yx = xy = (xy)y = (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 = hx, y, z : x2 = y2 = z3, z6 = 1, ((xαyβ)zγ )((xσyτ )zρ) = (xα+σyβ+τ )z3βσ+µγ+νρ, where µ and ν are defined as abovei.

Again, it is tedious to show that M is Moufang, but for the same reason as in Case 1(c1), M is not a group. The Moufang loop found here is denoted M24(G12,Q). Case 2. We now assume that M contains an element x of order 4, but no element of higher order. Assume also that M also contains an element y, which means that, from di-associativity, hx, yi is a group. If now y is of order ∼ 3, then we know from Table 4.2 that either |hx, yi| > 15 or hx, yi = C12 or ∼ G12. If |hx, yi| > 15, then |M| > 31 contrary to assumption, and if hx, yi = C12 or G12, then M contains an element of order greater than 4, also contrary to assumption. So M contains no element of order 3, and thus, apart from the identity element, M contains only elements of order 2 or 4. Since not every minimal set of generators contain an element of order 2, there exist, by Corollary 4.3, x, y, z ∈ M, all of order 4 such that hx, y, zi = M. We use now a result from Glauberman, Wright [5] implying that if each element of a Moufang loop is of order a power of 2, we have |M| = 2k for some k. Since |M| ≤ 31 we have |M| ≤ 16 and thus

16 ≥ |M| = |hx, y, zi| ≥ 2|hx, yi| ≥ 4|x| = 16 and so |M| = 16 and |hx, yi| = 8. ∼ 2 From Table 4.2, we thus have hx, yi = C2 × C4 or Q. Either way, x = y2, and, by analogous reasoning x2 = y2 = z2. Every element in M can be expressed as (xαyβ)zγ where α, β ∈ {0, 1} and 0 ≤ γ ≤ 3. Since |M| = 16 and there are 16 elements of this form and if (xa1 yb1 )zc1 = (xa2 yb2 )zc3 then c1−c2 a2 b2 a1 b1 −1 c1 c2 z = (x y )(x y ) ∈ hx, yi and thus c1 = c2, z = z and also xa1 = xa2 , yb1 = yb2 , each element of this form is therefore distinct. We need now determine how to multiply two elements of this form. ∼ For any w ∈ M, by Table 4.2 we have hw, xi = C2 × C4,D4,Q or C4. In any 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 hx, yi = C2 × C4 or Q and this also holds for hx, zi and hy, zi. There are thus four cases that need to be considered: ∼ ∼ ∼ Case 2(a) hx, yi = hx, zi = hy, zi = C2 × C4 ∼ ∼ ∼ Case 2(b) hx, yi = Q, hx, zi = hy, zi = C2 × C4 ∼ ∼ ∼ Case 2(c) hx, yi = C2 × C4, hx, zi = hy, zi = Q

38 Case 2(d) hx, yi =∼ hx, zi =∼ hy, zi =∼ Q In 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)

2 using that M is Moufang, that z is in the center of M and that C2 × C4 is abelian. Now we know, by Moufang’s theorem, that x(yz) 6= (xy)z since M is not a group. Thus, z(xy) 6= (xy)z. ∼ In Case 2(a), |xy| = 2, so we have that hxy, zi = C2 × C4 or D4. But since ∼ −1 z(xy) 6= (xy)z we must have hxy, zi = D4 and thus z(xy) = (xy)z . ∼ Similarly, in case 2(b), |xy| = 4 so hxy, zi = C2 × C4 or Q. Again, z(xy) 6= (xy)z so hxy, zi =∼ 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 all other elements in M. Also zy = yz and zx = xz. Thus in cases 2(a) and 2(b), ( (xδyε)z−η if δ ≡ ε ≡ 1 (mod 2) zη(xδyε) = (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+τ zνρ. due to a Moufang identity, alternativity, the fact that x2αy2β is in the center and 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 = hx, y, z : x2 = y2 = z2; z4 = 1; (xαyβ)zγ · (xσyτ )zρ = (xα+σyβ+τ )zµγ+νρ, where µ = (−1)στ and ν = (−1)ατ+βσi

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

39 In case 2(b), we have yx = xy3 = (xy)y2 = (xy)z2, and z2 is in the center of 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 is straightforward 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 = hx, 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 hx, zi =∼ Q, so |xz| = 4 and hx, xzi =∼ Q. As in 2(a), ∼ hxy, zi = 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 hy, xzi. 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 hy, xzi = C2 × C4. Choosing x, xz, y as our set ∼ ∼ ∼ of generators for a loop M, we have hx, xzi = Q, hx, yi = hxz, yi = C2 × C4. We see 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 ( (xδyε)zη if δ ≡ ε ≡ 2 (mod 2) zη(xδyε) = (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 = hx, 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 also assuming that not all minimal sets of generators of M contain an element of order 2, M must contain a minimal set of generators such that each element in the set is of order 3. From Lemma 4.1 and its corollaries, such a set must contain 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 hg1, g2i = ∼ A4 or C3 ×C3. We assume that x∈ / hg1, g2i. If hg1, g2i = A4, then, by Corollary ∼ 4.2, |hg1, g2, xi| ≥ |A4||x| = 36, contrary to assumption. So hg1, g2i = C3 × C3 and whichever elements g1 and g2 are, we have |g1g2| = 3. So K has exponent 3, meaning that for any element a ∈ K, a3 = e, and since x, y, z ∈ K we have K = 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) −1 e y n y x12 x3 x6 x9 x1 x10 x7 x4 x2 x11 x8 x5 −1 y n y e x9 x12 x3 x6 x10 x7 x4 x1 x11 x8 x5 x2 −1 n y e y x6 x9 x12 x3 x7 x4 x1 x10 x8 x5 x2 x11 −1 y e y n x3 x6 x9 x12 x4 x1 x10 x7 x5 x2 x11 x8 −1 x12 x3 x6 x9 e y n y x2 x5 x8 x11 x1 x4 x7 x10 −1 x3 x6 x9 x12 y e y n x11 x2 x5 x8 x10 x1 x4 x7 −1 x6 x9 x12 x3 n y e y x8 x11 x2 x5 x7 x10 x1 x4 −1 x9 x12 x3 x6 y n y e x5 x8 x11 x2 x4 x7 x10 x1 −1 x1 x4 x7 x10 x2 x11 x8 x5 e y n y x12 x3 x6 x9 −1 x10 x1 x4 x7 x5 x2 x11 x8 y e y n x3 x6 x9 x12 −1 x7 x10 x1 x4 x8 x5 x2 x11 n y e y x6 x9 x12 x3 −1 x4 x7 x10 x1 x11 x8 x5 x2 y n y e x9 x12 x3 x6 −1 x2 x5 x8 x11 x1 x10 x7 x4 x12 x3 x6 x9 e y n y −1 x11 x2 x5 x8 x4 x1 x10 x7 x3 x6 x9 x12 y e y n −1 x8 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 n y e

Table 6.2: Cayley table of M16(Q, 2) −1 −1 −1 e y1 n y1 y2 y3 y2 y3 x1 x3 x5 x7 x2 x4 x6 x8 −1 −1 −1 y1 n y1 e y3 y2 y3 y2 x7 x1 x3 x5 x4 x6 x8 x2 −1 −1 −1 n y1 e y1 y2 y3 y2 y3 x5 x7 x1 x3 x6 x8 x2 x4 −1 −1 −1 y1 e y1 n y3 y2 y3 y2 x3 x5 x7 x1 x8 x2 x4 x6 −1 −1 −1 y2 y3 y2 y3 n y1 e y1 x6 x8 x2 x4 x1 x3 x5 x7 −1 −1 −1 y3 y2 y3 y2 y1 n y1 e x4 x6 x8 x2 x3 x5 x7 x1 −1 −1 −1 y2 y3 y2 y3 e y1 n y1 x2 x4 x6 x8 x5 x7 x1 x3 −1 −1 −1 y3 y2 y3 y2 y1 e y1 n x8 x2 x4 x6 x7 x1 x3 x5 −1 −1 −1 x1 x3 x5 x7 x2 x8 x6 x4 e y1 n y1 y2 y3 y2 y3 −1 −1 −1 x3 x5 x7 x1 x4 x2 x8 x6 y1 e y1 n y3 y2 y3 y2 −1 −1 −1 x5 x7 x1 x3 x6 x4 x2 x8 n y1 e y1 y2 y3 y2 y3 −1 −1 −1 x7 x1 x3 x5 x8 x6 x4 x2 y1 n y1 e y3 y2 y3 y2 −1 −1 −1 x2 x8 x6 x4 x5 x7 x1 x3 y2 y3 y2 y3 e y1 n y1 −1 −1 −1 x4 x2 x8 x6 x7 x1 x3 x5 y3 y2 y3 y2 y1 e y1 n −1 −1 −1 x6 x4 x2 x8 x1 x3 x5 x7 y2 y3 y2 y3 n y1 e y1 −1 −1 −1 x8 x6 x4 x2 x3 x5 x7 x1 y3 y2 y3 y2 y1 n y1 e

42 Table 6.3: Cayley table of M16(C2 × C4) −1 −1 −1 0 0 e y3 n y3 y1 x3 y1 x6 y2 x5 y2 x2 x1 x7 x4 x8 −1 −1 −1 0 0 y3 n y3 e x3 y1 x6 y1 x5 y2 x2 y2 x8 x1 x7 x4 −1 −1 −1 0 0 n y3 e y3 y1 x6 y1 x3 y2 x2 y2 x5 x4 x8 x1 x7 −1 −1 −1 0 0 y3 e y3 n x6 y1 x3 y1 x2 y2 x5 y2 x7 x4 x8 x1 −1 −1 0 0 −1 y1 x3 y1 x6 n y3 e y3 x4 x7 x1 x8 y2 x2 y2 x5 −1 −1 0 0 −1 x3 y1 x6 y1 y3 e y3 n x8 x4 x7 x1 x2 y2 x5 y2 −1 −1 0 0 −1 y1 x6 y1 x3 e y3 n y3 x1 x8 x4 x7 y2 x5 y2 x2 −1 −1 0 0 −1 x6 y1 x3 y1 y3 n y3 e x7 x1 x8 x4 x5 y2 x2 y2 −1 0 0 −1 −1 y2 x5 y2 x2 x4 x7 x1 x8 n y3 e y3 y1 x6 y1 x3 −1 0 0 −1 −1 x5 y2 x2 y2 x8 x4 x7 x1 y3 e y3 n x6 y1 x3 y1 −1 0 0 −1 −1 y2 x2 y2 x5 x1 x8 x4 x7 e y3 n y3 y1 x3 y1 x6 −1 0 0 −1 −1 x2 y2 x5 y2 x7 x1 x8 x4 y3 n y3 e x3 y1 x6 y1 0 0 −1 −1 −1 x1 x7 x4 x8 y2 x2 y2 x5 y1 x6 y1 x3 e y3 n y3 0 0 −1 −1 −1 x7 x4 x8 x1 x5 y2 x2 y2 x3 y1 x6 y1 y3 e y3 n 0 0 −1 −1 −1 x4 x8 x1 x7 y2 x5 y2 x2 y1 x3 y1 x6 n y3 e y3 0 0 −1 −1 −1 x8 x1 x7 x4 x2 y2 x5 y2 x6 y1 x3 y1 y3 n y3 e

Table 6.4: Cayley table of M16(C2 × C4,Q) −1 −1 −1 −1 −1 e n x1 x3 x2 x4 y5 y5 y1 y1 y2 y2 y3 y3 y4 y4 −1 −1 −1 −1 −1 n e x3 x1 x4 x2 y5 y5 y1 y1 y2 y2 y3 y3 y4 y4 −1 −1 −1 −1 −1 x1 x3 e n y5 y5 x2 x4 y2 y2 y1 y1 y4 y4 y3 y3 −1 −1 −1 −1 −1 x3 x1 n e y5 y5 x4 x2 y2 y2 y1 y1 y4 y4 y3 y3 −1 −1 −1 −1 −1 x2 x4 y5 y5 e n x3 x1 y4 y4 y3 y3 y2 y2 y1 y1 −1 −1 −1 −1 −1 x4 x2 y5 y5 n e x1 x3 y4 y4 y3 y3 y2 y2 y1 y1 −1 −1 −1 −1 −1 y5 y5 x4 x2 x1 x3 n e y3 y3 y4 y4 y1 y1 y2 y2 −1 −1 −1 −1 −1 y5 y5 x2 x4 x3 x1 e n y3 y3 y4 y4 y1 y1 y2 y2 −1 −1 −1 −1 −1 y1 y1 y2 y2 y4 y4 y3 y3 n e x1 x3 y5 y5 x4 x2 −1 −1 −1 −1 −1 y1 y1 y2 y2 y4 y4 y3 y3 e n x3 x1 y5 y5 x2 x4 −1 −1 −1 −1 −1 y2 y2 y1 y1 y3 y3 y4 y4 x1 x3 n e x2 x4 y5 y5 −1 −1 −1 −1 −1 y2 y2 y1 y1 y3 y3 y4 y4 x3 x1 e n x4 x2 y5 y5 −1 −1 −1 −1 −1 y3 y3 y4 y4 y2 y2 y1 y1 y5 y5 x2 x4 n e x1 x3 −1 −1 −1 −1 −1 y3 y3 y4 y4 y2 y2 y1 y1 y5 y5 x4 x2 e n x3 x1 −1 −1 −1 −1 −1 y4 y4 y3 y3 y1 y1 y2 y2 x4 x2 y5 y5 x1 x3 n e −1 −1 −1 −1 −1 y4 y4 y3 y3 y1 y1 y2 y2 x2 x4 y5 y5 x3 x1 e n

43 Table 6.5: Cayley table of M16(Q) −1 −1 −1 −1 −1 −1 −1 e n y1 y1 y2 y2 y3 y3 y4 y4 y5 y5 y6 y6 y7 y7 −1 −1 −1 −1 −1 −1 −1 n e y1 y1 y2 y2 y3 y3 y4 y4 y5 y5 y6 y6 y7 y7 −1 −1 −1 −1 −1 −1 −1 y1 y1 n e y3 y3 y2 y2 y7 y7 y6 y6 y5 y5 y4 y4 −1 −1 −1 −1 −1 −1 −1 y1 y1 e n y3 y3 y2 y2 y7 y7 y6 y6 y5 y5 y4 y4 −1 −1 −1 −1 −1 −1 −1 y2 y2 y3 y3 n e y1 y1 y6 y6 y7 y7 y4 y4 y5 y5 −1 −1 −1 −1 −1 −1 −1 y2 y2 y3 y3 e n y1 y1 y6 y6 y7 y7 y4 y4 y5 y5 −1 −1 −1 −1 −1 −1 −1 y3 y3 y2 y2 y1 y1 n e y5 y5 y4 y4 y7 y7 y6 y6 −1 −1 −1 −1 −1 −1 −1 y3 y3 y2 y2 y1 y1 e n y5 y5 y4 y4 y7 y7 y6 y6 −1 −1 −1 −1 −1 −1 −1 y4 y4 y7 y7 y6 y6 y5 y5 n e y3 y3 y2 y2 y1 y1 −1 −1 −1 −1 −1 −1 −1 y4 y4 y7 y7 y6 y6 y5 y5 e n y3 y3 y2 y2 y1 y1 −1 −1 −1 −1 −1 −1 −1 y5 y5 y6 y6 y7 y7 y4 y4 y3 y3 n e y1 y1 y2 y2 −1 −1 −1 −1 −1 −1 −1 y5 y5 y6 y6 y7 y7 y4 y4 y3 y3 e n y1 y1 y2 y2 −1 −1 −1 −1 −1 −1 −1 y6 y6 y5 y5 y4 y4 y7 y7 y2 y2 y1 y1 n e y3 y3 −1 −1 −1 −1 −1 −1 −1 y6 y6 y5 y5 y4 y4 y7 y7 y2 y2 y1 y1 e n y3 y3 −1 −1 −1 −1 −1 −1 −1 y7 y7 y4 y4 y5 y5 y6 y6 y1 y1 y2 y2 y3 y3 n e −1 −1 −1 −1 −1 −1 −1 y7 y7 y4 y4 y5 y5 y6 y6 y1 y1 y2 y2 y3 y3 e n

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