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Moufang Loops General Theory and Visualization of Non-Associative Moufang Loops of Order 16

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Moufang Loops General Theory and Visualization of Non-Associative Moufang Loops of Order 16 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 jGj 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 2 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 2 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.
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