A STUDY ON THE ALGEBRAIC STRUCTURE OF SL2 Z pZ ( ~ ) A Thesis Presented to The Honors Tutorial College Ohio University In Partial Fulfillment of the Requirements for Graduation from the Honors Tutorial College with the degree of Bachelor of Science in Mathematics by Evan North April 2015 Contents 1 Introduction 1 2 Background 5 2.1 Group Theory . 5 2.2 Linear Algebra . 14 2.3 Matrix Group SL2 R Over a Ring . 22 ( ) 3 Conjugacy Classes of Matrix Groups 26 3.1 Order of the Matrix Groups . 26 3.2 Conjugacy Classes of GL2 Fp ....................... 28 3.2.1 Linear Case . .( . .) . 29 3.2.2 First Quadratic Case . 29 3.2.3 Second Quadratic Case . 30 3.2.4 Third Quadratic Case . 31 3.2.5 Classes in SL2 Fp ......................... 33 3.3 Splitting of Classes of(SL)2 Fp ....................... 35 3.4 Results of SL2 Fp ..............................( ) 40 ( ) 2 4 Toward Lifting to SL2 Z p Z 41 4.1 Reduction mod p ...............................( ~ ) 42 4.2 Exploring the Kernel . 43 i 4.3 Generalizing to SL2 Z p Z ........................ 46 ( ~ ) 5 Closing Remarks 48 5.1 Future Work . 48 5.2 Conclusion . 48 1 Introduction Symmetries are one of the most widely-known examples of pure mathematics. Symmetry is when an object can be rotated, flipped, or otherwise transformed in such a way that its appearance remains the same. Basic geometric figures should create familiar examples, take for instance the triangle. Figure 1: The symmetries of a triangle: 3 reflections, 2 rotations. The red lines represent the reflection symmetries, where the trianlge is flipped over, while the arrows represent the rotational symmetry of the triangle. These notions can be expanded to any regular polygon, certain three-dimensional ob- jects, and even certain objects in nature such as crystal formations. Objects that possess a large number of symmetries are regarded as beautiful and intriguing, of- ten invoking further studies into the object's special properties. However, physical shapes are not the only objects that can possess symmetry, and the formalization of this concept has come to be known as group theory. 1 Group theory is a branch of abstract algebra that is largely concerned with the study of symmetries of mathematical objects. A group is a mathematical abstraction of symmetries, and using group theory, mathematicians find hidden patterns in the symmetries of a problem. A prime example of this is Galois's proof of the unsolvability of a quintic equation, which uses symmetries of the roots of a polynomial. Even though group theory often focuses on concepts more abstract than geo- metric objects, it can still be used to study familiar shapes. In this thesis, special focus is given to groups related to symmetries of the torus. The torus is a very rec- ognizable shape, even outside the mathematical world, with many people knowing it as a doughnut, a bagel, or even an inner-tube. Whatever your impression of the torus is, however, it should resemble figure 2. Figure 2: A torus. Accessed from [Mrabet, 2007] 2 The torus appears in many areas of mathematics. Mathematically, the easiest 2 way to view the torus is as the real plane modded by its integer points, R 2. Z This process can be viewed as taking a plane, rolling it into a tube (thus moddingÒ out one copy of Z), then connecting the ends of that tube to mod out the other copy of Z. This can be visualized in figure 3. Figure 3: How to make a torus from the unit square. From [Kostyuk, 2009] The rigid symmetries of the torus form what is called the modular group, or SL2 Z . Elements of the modular group are size and orientation preserving linear transformations,( ) otherwise known as 2 2 determinant one matrices, with integer coefficients. × 3 This paper looks to shed light on the structure of finite quotients of the modular group, SL2 Z NZ for N N. This is approached by breaking N down into its prime components,( ~ ) then considering∈ the corresponding decomposition of the finite quotient of the modular group. r mi N pi i=1 r = M mi SL2 Z NZ SL2 Z pi Z i=1 ( ~ ) ≃ ( ~ ) The simplest of these cases is obviously when the power of prime p is one, which is our base case as well as our main case of study. The main topic of this paper is conjugacy classes of the groups of interest. Conjugacy is an equivalence relation on groups, so looking at conjugacy classes is one way to see the hidden structure. Once the structure of the conjugacy classes of SL2 Z pZ is established, some preliminary results are discussed for extending findings( ~ to higher) powers. It should be stated now that the results on SL2 Z pZ have been known for some time, and are verifiable in [Humphreys, 1975], [Reeder,( ~ ) 2008], and [Dornhoff, 1971]. However, these sources are written much more compactly, often omitting highly relevant statements or their proofs. Most of the results in chapter 2 are proven, but great care has been taken in chapter 3 to give not only all of the relevant proofs, but to explain the material in such a way that the underlying structure of the conjugacy classes is more easily visualized. 4 2 Background 2.1 Group Theory This thesis is ultimately a study in group theory, so there are some fundamentals that need to be reviewed in order to ensure the reader has any necessary definitions and background available. This section presents all of the definitions, theorems, and lemmas that are needed to approach the later content, given with proof when deemed appropriate. Throughout the section, groups will be assumed to be finite, so special cases for infinite groups will not be considered. Finally, any of the definitions and results in this section can be found in any introductory abstract algebra book. For this paper, they have been adapted from [Dummit and Foote, 2004] to suit the needs of this review. Definition 1. A group is a set G together with an operation G G G that satisfies the following properties: ⋅ ∶ × → (1) The operation is associative: f g h f g h g; f; h G (2) There exists a⋅ unique identity element⋅ ( ⋅ )e:= e( g⋅ )g⋅ ∀g e g ∈ G (3) For every g G, there exists a unique inverse⋅ element= = ⋅g−∀1 that∈ satisfies g g−1 e g−1 g ∈ ⋅ = = ⋅ Groups are sets with a special structure imposed by their associated operation, and this structure gives rise to a surprising number of properties. The next few definitions provide the language to discuss and derive many of these properties. Additionally, throughout this paper, the operation of a group will commonly be omitted. It is popular shorthand to set g h gh, as the operation is typically either implied or irrelevant (from abstraction).⋅ = One thing to note here is that the operation is not necessarily commutative, meaning that in general, gh hg for ≠ 5 two element g; h G. However, there are plenty of examples where this does hold, such as the integers∈ with addition. In fact, this is a common (and notable) enough property that these groups have their own name. Definition 2. An abelian group is a commutative group, i.e. for every g; h G, gh hg. ∈ =The most familiar example of an abelian group should be the integers, Z, paired with addition. From grade school we know addition is both associative and commutative, and zero plus anything is zero, so zero serves as the identity. Negative numbers are additive inverses, so Z satisfies the definition of an abelian group. Another interesting example to consider is the modular integers, Z nZ. Both Z and Z nZ are classic examples of what are called cyclic groups. ~ Definition 3.~ A cyclic group is a group that is generated by one element, i.e. there exists some g G such that for every h G, there exists an nh such that gnh h. In this case,∈g is called a generator of∈ G. In= a cyclic group, repeatedly composing a generator with itself will eventually return the entire group. One especially nice thing about cyclic groups is that they are always abelian, so if one can find a single generator of a group then it must be abelian. Definition 4. The order of a group G, denoted G , is the number of elements in G. The order of an element g G, denotedS S g is the smallest positive integer n such that gn e, where gn ∈denotes n repeatedO( ) compositions of g with itself. = Definition 5. H is a subgroup of G if H is a subset of G, and H satisfies the properties from definition 1 with respect to the operation in G. In symbols, this is written H G. ≤ 6 Definition 6. For a subgroup H G and g G, gH is the left coset of H with respect to g and Hg is the right coset≤ of H∈with respect to g, where gH gh h H Hg hg h H = { S ∈ } = { S ∈ } Subgroups are simply subsets that also have the group structure. Because of the properties of groups, being a subgroup has close ties to the divisibility of the order of groups and their subgroups. One can also consider what happens to all the elements of a subgroup when composed with other elements of the group, creating cosets. The only difference between a left and right coset is what side the elements are composed from, a subtle but necessary distinction because not all groups are commutative (i.e.