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A STUDY ON THE 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

by Evan North April 2015 Contents

1 Introduction 1

2 Background 5 2.1 Theory ...... 5 2.2 Linear ...... 14

2.3 Group SL2 R Over a ...... 22

( ) 3 Conjugacy Classes of Matrix Groups 26 3.1 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 ...... 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. 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 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 of the triangle. These notions can be expanded to any regular , 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 .

1 Group theory is a branch of 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 . 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 . 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 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 . From [Kostyuk, 2009]

The rigid symmetries of the torus form what is called the , or

SL2 Z . Elements of the modular group are size and orientation preserving linear

transformations,( ) otherwise known as 2 2 one matrices, with integer coefficients. ×

3 This paper looks to shed light on the structure of finite 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 of the modular group.

r mi N 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 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 G together with an 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 ⋅ ( ⋅ )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 ∈ with . In fact, this is a common (and notable) enough property that these groups have their own name.

Definition 2. An 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 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 of G if H is a 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 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 ∈ } are simply 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 . 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. gh hg). However, there are certain subgroups whose structure is ”nice enough” to create≠ this illusion of commutativity, and they are given the distinction ”normal”.

Definition 7. For a subgroup N G, N is a of G if for every g G, gN Ng. A normal subgroup≤ is denoted N G. ∈ = ◁ In a normal subgroup N, left cosets with respect to g coincide with right cosets with respect to g. Note that this does not necessarily make the elements of G commutative, but is rather a property of the subgroup itself. By multiplying with g−1 from the right on each side of the equation, the condition can be rewritten as gNg−1 N, a useful operation known as conjugation. In this situation, it is often said that= ”N is invariant under conjugation”.

7 Definition 8. Conjugation is a relation such that g f if and only if there

exists some h such that hgh−1 f. ∼ ∼ = The operation of conjugation will allow us to introduce important concepts and will be used throughout this thesis. A quick check can verify that conjugation is in fact an equivalence relation.

Proposition 1. Conjugation satisfies the three properties of an equivalence rela- tion.

Proof. The first property is the easiest. To show g g, conjugate g with the

identity, e, to get ege−1 which is obviously just g. Assume∼ g f. Then there exists some h such that hgh−1 f. After composing with h−1 on∼ the left and h on the right, this becomes g h−1=fh, implying f g. Lastly we check the transitive

−1 property. Let f g and g = h. Then there exist∼x1 and x2 such that x1fx1 g and

−1 −1 −1 x2gx2 h. Substitutions∼ ∼ and rearrangements then yield that x2x1fx1 x2 = h, so f h. = =

∼Note that conjugating an element g with an element that commutes with it will never change it because the conjugating element will simply pass through to its inverse and reduce to the identity, so in an abelian group every subgroup is normal. The notion of conjugation leads to the definition of one of the central structures in this study.

Definition 9. For any g G, the G- of g is the set of all elements

of G that can be obtained∈ through conjugation by other elements of G

gG xgx−1 x G

= ™ T ∈ ž

8 For a given group, conjugacy classes can be computed by explicitly conjugating one g G with every element of G. However, this can be long, tedious work, espe- cially for∈ large enough groups. Instead, focus is placed on finding similar properties between elements that would allow us to separate elements into conjugacy classes. It is common practice to select one element from a conjugacy class to serve as a representative for the rest of the class, which will be seen later in chapter 3.

To keep with the earlier example, consider conjugacy classes of Z nZ. Note that if you pick one number x Z nZ, and you add y to the left and~ subtract y from the right, you will just get ∈x back.~ This means the conjugacy class of x is just itself as a single element. This result actually holds in general for commutative elements of groups. If an element commutes with the rest of the group, then its conjugacy class is exactly itself. The patterns that arise in the discussion of conjugacy classes allow comparisons between groups, such as how different groups can have similar structure. The formal way to compare groups is done by means of special maps between them called .

Definition 10. Let G, and H, be groups. A group is a map φ G H from( a group⋆) G (to a⋅) group H that satisfies the property ∶ → φ g h φ g φ h g, h G

( ⋆ ) = ( ) ⋅ ( ) ∀ ∈ From definition 10, several useful properties of group homomorphisms can quickly be derived, such as φ e e and φ g−1 φ g −1. Group homomorphisms are structure-preserving maps( from) = one group( ) to= another,( ) establishing similarities between the structures of two groups. This can be taken one step further by es- tablishing a special class of homomorphisms that capture what it means for two

9 groups to be ”identical”.

Definition 11. A φ G H is a group if there exists a group homomorphism φ−1 H ∶ G→such that φ φ−1 g g for every g G. ∶ → ( ( )) = ∈ Two groups that are isomorphic to one another share the same internal struc- ture. Sometimes problems are easier to solve or understand when written in terms of a group that is isomorphic to the group of interest. This idea of swapping out groups as a matter of convenience makes the inverse notation used in definition 11 that much more appropriate. Having an isomorphism φ means there is auto- matically a homomorphism that ”undoes” φ, a trait commonly known as being an inverse. can be invaluable in determining unknown properties of a group if that groups can be shown to be isomorphic to a more familiar group. Often, only a few properties of a group are needed to determine if it is isomorphic to another group. The next two definitions describe two ways to ”manufacture” new groups from known ones. Intuitively, the first can be thought of as a way to ”multiply” two groups, and the second is a sort of ”” for groups. However, these are extremely loose descriptions, and the reality of what happens in these actions is far more nuanced. For now, only the second will receive focus, but the other will come back up later.

Definition 12. The direct of groups G and H is pairs of g G and h H such that g1, h1 , h2 g1 g2, h1 h2 . The is∈ denoted G∈ H. If G and (H are) abelian⋅ ( groups,) = ( then⋅ this⋅ action) is referred to as the × of G and H, or G H. ⊕

10 G Definition 13. G and H G, the H is the set of left cosets of the group H ◁ Ò G H gH g G Ò = { S ∈ } This concept is referred to as ”modding G by H”. The idea of modding one group by another is useful in making given groups into smaller groups with similar structure, and is a notion used several times throughout this thesis. A quotient G group H can be viewed as G with an equivalence relation established by H. This meansÒ that elements in G are considered equal if they only differ by a factor of H, which can be written as g1 g2 if and only if there exists some h H such that g1 g2h. The next theorem∼ describes finding the size of a given∈ quotient group. =

Theorem 1 (Lagrange). If G is a finite group and H is a subgroup of G, then

G G H H S S S Ò S = S S G Proof. From definition 13, we know every element of H is a left coset gH. Each of these cosets will have one element for each elementÒ of H, so each coset has exactly the same size. Cosets are either disjoint or the same, so the order of the group must be the size of the cosets times the number of cosets. Therefore, the order of the quotient group equals the quotient of the orders of the groups themselves.

11 Definition 14. The kernel of a group homomorphism φ G H is the subset of

G that maps to the identity in H ∶ →

ker φ g G φ g e

( ) = { ∈ S ( ) = } Lemma 1. The kernel of a group homomorphism φ G H is a normal subgroup of G. ∶ →

Proof. Let K ker φ and take any g G. Consider gKg−1 gkg−1 k K .

Note that from= the( definition) of homomorphisms,∈ φ gkg−1 = {φ g φ kS φ∈g−1}. From the followup to definition 10, the inverse term can( be rewritten) = ( ) as( φ) g(−1 ) φ g −1. From the definition of the kernel, φ k e. Therefore, φ gkg( −1 ) = φ(g)φ k φ g−1 φ g eφ g −1 e, so gkg−1 will( ) be= in K for every (g G,) so= gKg( )−1( )K(for) every= (g ) G(. ) = ∈

The= kernel is an important∈ invariant of a homomorphism, and as such it plays a central role in chapters 3 and 4. One useful property is displayed in lemma 1. According to this lemma, if a homomorphism can be constructed from G into H with N as its kernel, then N is normal in G. Now that quotient groups and kernels of group homomorphisms have been introduced, there is an important theorem that ties these concepts together.

Theorem 2. Let G and H be groups, and let φ G H be a surjective group G homomorphism. Then the quotient group ker φ∶ is→ isomorphic to H. Ò This is just the First Isomorphism Theorem, an( integral) theorem in any abstract algebra course. This result will be cited several times throughout this paper, so readers are advised to be comfortable with its statement.

12 Definition 15. The centralizer of an element g G is the set

∈ −1 CG g x G xgx g

( ) = ™ ∈ T = ž The centralizer of an element is the set of elements that do not change g under conjugation, which is equivalent to saying they commute with g because one each side of the equation can be multiplied by x to get xg gx. The centralizer of g G

is a subset of G, and in an abelian group the centralizer= is always the whole group.∈ Having access to the centralizer allows for one more theorem in this section, which relates the size of the centralizer to the size of conjugacy classes in G.

Theorem 3 (Counting Formula). For any g G, the order of G is equal to the

order of the G-conjugacy class of g times the size∈ of the G-centralizer of g, i.e.

G G g CG g

S S = S S ⋅ S ( )S Proof. This proof begins by considering a map ϕ G gG. Given an CG g element f G, ϕ fC g fgf −1. Note that this means if fC g hC g , G ∶ Ò ( ) →G G −1 −1 then fgf ∈ hgh( , which( )) puts↦ left cosets of CG g in a one to one correspondence( ) = ( ) with elements= of gG. This means that gG is equal( ) to the number of left cosets of SGS CG g , which is equal to . Multiplying by CG g returns the formula from SCG(g)S S S the( theorem.) S ( )S

The advantage to using this formula is that if the centralizer of one element can be computed, then the size of that element’s conjugacy class is known. The last definition in this section introduces a concept that, on an abstract level, lies at the heart of what this study is about. However, its technical details and properties will not be noted or referenced until the of the study.

13 Definition 16. An exact is a sequence of groups and homomorphisms between them such that the of each homomorphism is just the kernel of the next. A common example is

ϕ 1 ker ϕ G H 1

→ ( ) → Ð→ → Exact create a concise description of the related structure of groups, and are useful for classifying different kinds of problems.

2.2

Much like section 2.1, this section will be largely review, and can be skimmed or altogether skipped by someone comfortable with the results of a typical un- dergraduate linear algebra course. One thing to note about this section is that while many definitions and results have analogues for higher dimensional cases (i.e. generic n n matrices), primary focus is given to the 2 2 case. Several re- sults become much× easier when explicitly making calculations× and therefore many results will not be generalized. The ultimate goal of this section is to provide all the necessary background to comfortably use canonical forms of matrices.

Definition 17. A ring is a set R with two operations, call them and , that satisfy the following properties (with a, b, c R): + ⋅ (1) R, is an abelian group. ∈ (2) ( is+ associative.) (3)⋅ Left distributivity: a b c a b a c (4) Right distributivity: ⋅ a( +b ) c= a⋅ +c b⋅ c ( + ) ⋅ = ⋅ + ⋅

14 From part (1) of this definition, it is clear that rings are just groups (specifically abelian groups) with extra structure imposed on their elements. This structure comes in the form of a second operation that behaves in a particularly nice way with the first operation through left and right distributivity. A major difference to note right away is the absence of an identity and inverses for this second operation, elements that have to exist in a group. When an identity is present, the ring is commonly referred to as a unital ring, but for our purposes rings will always be assumed to be unital. Inverses are a little bit trickier when dealing with the second operation. If inverses are present for the ring, we only require that they exist for ”non-zero” elements, where zero is the identity of .

+ Definition 18. A is a ring in which every element is multi- plicatively commutative.

Matrices can be defined over any ring, but here the focus will be on matrices defined over commutative rings. Therefore, it will be assumed that all matrices in this section are declared over a commutative ring. If a matrix is declared without specifying its coefficients, random letters will be assigned with the intent that their origin is clear from the context or otherwise do not affect the structure of the statement.

Definition 19. Matrix product of matrices M and N is defined by

a b a′ b′ aa′ bc′ ab′ bd′ MN ⎛c d⎞ ⎛c′ d′⎞ ⎛ca′ +dc′ cb′ +dd′⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ + + ⎠

15 Definition 20. The is the matrix I that when multiplied by other matrices, returns the other matrix, i.e. MI IM M. The 2 2 form is

= = × 1 0 I ⎛0 1⎞ ⎜ ⎟ = ⎜ ⎟ ⎝ ⎠ Definition 21. For matrices over a ring R, a scalar matrix is a formed by multiplying the identity matrix by a ring element

1 0 r 0 r ⎛0 1⎞ ⎛0 r⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ Proposition 2. Scalar matrices commute with all matrices.

Proof. First, write the scalar matrix as the ring element r R, a commutative ring, times the identity. Then the ring element will be able∈ to pass through the other matrix (because it commutes with all the coefficients). All that’s left is the identity matrix and the other matrix, which will simplify to just the other matrix. This can be written formulaically as

r 0 a b 1 0 a b a b r r ⎛0 r⎞ ⎛c d⎞ ⎛0 1⎞ ⎛c d⎞ ⎛c d⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ ⎝a b⎠ ⎝r 0⎠ ⎝a b⎠ ⎝1 0⎠ ⎝a b⎠ r r ⎛c d⎞ ⎛0 r⎞ ⎛c d⎞ ⎛0 1⎞ ⎛c d⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

16 Definition 22. The determinant (abbreviated det) of a 2 2 matrix is defined as follows × a b det ad bc ⎛c d⎞ ⎜ ⎟ ⎜ ⎟ = − Lemma 2. The determinant of a⎝ product⎠ is the product of , i.e. for two matrices M and N,

det MN det M det N

( ) = ( ) ( ) Proof. The proof proceeds by a direct calculation in the 2 2 case.

× a b a′ b′ aa′ bc′ ab′ bd′ det det ⎛⎛c d⎞ ⎛c′ d′⎞⎞ ⎛ca′ +dc′ cb′ +dd′⎞ ⎜⎜ ⎟ ⎜ ⎟⎟ ⎜ ⎟ ⎜⎜ ⎟ ⎜ ⎟⎟ = ⎜ ⎟ ⎝⎝ ⎠ ⎝ ⎠⎠ aa⎝′ bc+′ cb′ dd+′ ⎠ab′ bd′ ca′ dc′

= (aca′b+′ ada)( ′d′+ bcb)′c−′ ( bdc+′d′ )( + )

= aca′b+′ adb′c+′ bca′d+′ bdc′d′

−ada′d′ −bca′d′ −bcb′c′ −adb′c′

= ad bc− a′d′ +ad bc−b′c′

= (ad − bc) a′d′− (b′c′− )

= ( − )( − ) The final line is equivalent to the product of the individual determinants, so we are done.

For those who have seen matrix algebra before these should be very familiar formulas, and they are used extensively for calculations throughout the paper.

1 0 The identity matrix, denoted I, is the matrix 0 1 , and has the propery that its product with any matrix is just that matrix,( i.e.) MI IM M. Note here

17 = = that matrix addition has not been formally defined because it is simply performed -wise, leaving no special structure or commentary to be gleaned. There are systematic ways to define matrix for general dimensions, but the formula from definition 19 will suffice for this study.

Definition 23. The of a matrix is the sum of its diagonal elements. For a 2 2 matrix the trace is a b × tr a d ⎛c d⎞ ⎜ ⎟ ⎜ ⎟ = + Lemma 3. The trace of the sum of⎝ matrices⎠ is the sum of their traces, i.e.

a b a′ b′ a b a′ b′ tr tr tr ⎛⎛c d⎞ ⎛c′ d′⎞⎞ ⎛c d⎞ ⎛c′ d′⎞ ⎜⎜ ⎟ ⎜ ⎟⎟ ⎜ ⎟ ⎜ ⎟ ⎜⎜ ⎟ + ⎜ ⎟⎟ = ⎜ ⎟ + ⎜ ⎟ ⎝⎝ ⎠ ⎝ ⎠⎠ ⎝ ⎠ ⎝ ⎠ Proof. A simple calculation shows that the left side is a a′ d d′ and the right side is a d a′ d′ . Thanks to associativity and( commutativity+ )+( + ) of addition, these two( + equations) + ( + are) equal.

Lemma 2 shows that the determinant splits over multiplication, and lemma 3 shows that the trace behaves analogously over addition. The trace of a matrix is also invariant under conjugation, giving it obvious value for this study. A central idea in linear algebra is the fact that any linear transformation can be represented by a matrix. The determinant of a matrix carries all information about its associated linear transformation, such as the scaling and orientation of the transformation. The determinant plays a very important role in this study as it will directly determine which matrices are of interest, and thankfully it behaves very nicely with respect to , as displayed by lemma 2.

18 Definition 24. Two columns (rows) of a matrix are called linearly independent if one cannot be written as a multiple of the other. Otherwise, the two columns (rows) are called linearly dependent.

Theorem 4. The determinant of a matrix is non-zero if and only if the columns (or rows) of that matrix are linearly independent.

While the concept of is a large part of linear algebra, its technical points are less relevant to this discussion. Theorem 3 is only needed in the first part of chapter 3, after which point the results from group theory will suffice.

a b Definition 25. If M is a 2 2 matrix c d , then the inverse of M can be computed as × ( ) 1 d b M −1 det M ⎛ c− a ⎞ = ⎜ ⎟ ( ) ⎜ ⎟ This formula can be derived, but it requires⎝− several⎠ other properties that are otherwise irrelevant to this study. One advantage to viewing the inverse of a matrix in this form is that it immediately becomes clear that only matrices with non-zero determinant are invertible. Additionally, it can be seen by direct computation that the determinant is invariant with respect to conjugation,

det XMX−1 det X det M det X−1 det X X−1 det M det M

( ) = ( ) ( ) ( ) = ( ⋅ ) ( ) = ( ) Having the inverse formula readily available will also prove handy for several cal- culations in chapter 3.

19 Definition 26. The polynomial of a matrix M is equal to det M tI ,

( − ) a t b det a t d t bc t2 a d t ad bc ⎛ −c d t⎞ ⎜ ⎟ ⎜ ⎟ = ( − )( − ) − = − ( + ) + ( − ) ⎝ − ⎠ t2 tr M t det M

= − ( ) + ( ) The characteristic polynomial of a matrix relies on the trace and determinant of said matrix, which are both invariant with respect to conjugation. Therefore, the characteristic polynomial will also be invariant with respect to conjuation. It also leads to a closely related concept, the minimal polynomial.

Definition 27. The minimal polynomial of a matrix M is a monic polynomial t of least degree such that M 0.

P( ) P( ) = The minimal and characteristic always share the same set of roots, and as such the minimal polynomial will always divide the characteristic polyno- mial. It is common for the minimal polynomial and characteristic polynomial to coincide in matrices, but it is not always guaranteed. Take, for example, a multiple

a 0 of the identity matrix aI 0 a . Using the formula for the characteristic poly-

2 2 nomial from definition 26,=det( aI) tI a t . If 1 t a t , then clearly

1 aI 0. However, the first( degree− ) polynomial= ( − ) 2 Pt ( ) a= (t −also) satisfies this requirement.P ( ) = Because 2 t has the lower degree (andP ( ) it= is( linear,− ) so there cannot be an even lower one),P 2( t) must be the minimal polynomial. Now that minimal and characteristic polynomialsP ( ) have been introduced, the final topic of this section can be addressed: Canonical forms.

20 Definition 28. The canonical form of a 2 2 matrix over an arbitrary field will be one of the following matrices ×

a 0 a 1 a 0 0 b ⎛0 a⎞ ⎛0 a⎞ ⎛0 b⎞ ⎛1 a⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ These matrices have minimal polynomials t a , t a 2, t a t b , and t2 at b (irreducible), respectively, and every( minimal− ) ( − polynomial) ( − )( will− fall) into (one− of these− ) categories. They are in some sense the simplest matrices with their given minimal polynomials, and as such are useful as goto examples of matrices that exist for a given minimal polynomial. Take for instance the last minimal polynomial, t2 at b. This minimal polynomial has the same degree as its cor- responding characteristic− − polynomial, so they must coincide (from minimal poly- nomial dividing characteristic polynomial). From definition 26, we know the trace of the associated matrix is a and the determinant is b. There are many ways to construct such a matrix, but the easiest is to put zero− and a on the diagonal to satisfy the trace condition, then 1 and b in the other corners for the determi- nant, matching the canonical form given above. A similar process can be repeated to check the other three forms with the exception of scalar matrices, which were covered in the discussion of definition 27 Because minimal polynomials are invariant with respect to conjugation, any matrix with one of these minimal polynomials will be conjugate to the correspond- ing canonical form. Canonical forms create much easier calculations and provide concise representatives for the relevant information in a matrix. For example, the determinant of the third example from above is clearly ab, a conclusion that is not as clear in certain conjugations of the matrix. Canonical forms lay the groundwork for the work done in chapter 3.

21 2.3 Matrix Group SL2 R Over a Ring

The purpose of this section is to( provide) the last couple definitions and properties that do not quite fit into group theory or linear algebra. There is a small amount of general that still needs covered, and some results specifically regarding the Special will be proven. One thing to note about the ring theory results is that in many cases, these new definitions are simply extended analogues to parts of group theory. Therefore, certain terms will be given less attention now because they behave similarly enough to their group counterpart. For now, consider the next object in the hierarchy of algebraic structures.

Definition 29. A field is a commutative ring with a for every non-.

In a field, terms can be both additively and multiplicatively canceled out, with the familiar example of not being able to divide by zero. The next chapter focuses heavily on the use of finite fields, specifically the case of fields with prime order.

The easiest example of a finite field is the integers mod some Z pZ, denoted Fp. It can quickly be confirmed that both addition and multiplication~ are commutative, and the only element with no multiplicative inverse is the congruence class of 0. However, this only works when the integers are modded by a prime number. For example, consider Z p2Z. This ring has what are referred to as zero-

, non-zero numbers whose~ product is zero. If p Z p2Z is multiplied by itself, it comes to p2, but in this ring p2 is equal to zero,∈ so~p p 0. This means that Z p2Z is not a field, but simply a commutative ring, a nuance⋅ = that will play a large~ role later in this study.

22 Definition 30. For rings R and S, a is a map ψ R S that satisfies the following properties (a, b R): ∶ → (1) ψ a b ψ a ψ b ∈ (2) ψ(ab+ )ψ= a(ψ) b+ ( )

(3) ψ(1R) = 1S( ) ( ) ( ) = Here, 1R and 1S are the multiplicative identities of R and S, respectively. Ring homomorphisms are a direct analogue to group homomorphisms, still holding the property of being structure-preserving maps. In addition to homomorphisms, ring isomorphisms can be defined analogously to their group counterpart. Rings can also be ”manufactured” like groups through the use of a direct sum. Note here that direct sum notation will be used throughout because rings are abelian groups (see definition 12 and definition 17). R S is simply ordered pairs

where both addition and multiplication work component-wise.⊕ A common example of direct sums of rings appears in breaking down some Z NZ. First, consider the

mi prime factorization of N. This gives a base of numbers p~i with which to create the direct sum as follows:

n mi N pi i=1 n = M mi Z NZ Z pi Z i=1 Ô⇒ ~ ≃ ? ~ This result is very well-known, and is essentially an abstract algebra analogue to the fundamental theorem of , and thus we will simply take it for granted here. Now that the some basic ring theory results have been discussed, we can focus more on combining what we know of group and ring theory to formulate some

general statements about our main group of study, SL2 R . This will start with

23 ( ) the formal definition of said group.

Definition 31. The is the set of 2 2, determinant one matrices over a given ring R: ×

a b SL2 R a, b, c, d R; ad bc 1R ⎧ R ⎫ ⎪ ⎛c d⎞ R ⎪ ⎪ ⎜ ⎟ R ⎪ ( ) = ⎨ ⎜ ⎟ R ∈ − = ⎬ ⎪ R ⎪ ⎪ ⎝ ⎠ R ⎪ The Special Linear Group⎩ can beR defined over any commutative⎭ ring R. Its n n version can also be defined, but for this study we are only concerned with the× 2 2 case. Many nice properties of rings translate naturally over SL2 R , and the relevant× ones are covered in the next several results. ( )

Lemma 4. If ψ R S is a ring homomorphism, then ψ induces a group homo- φ SL2∶ R→ SL2 S by the map ∶ ( ) → ( ) a b ψ a ψ b φ ⎛c d⎞ ⎛ψ(c) ψ(d)⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ↦ ⎜ ⎟ ⎝ ⎠ ⎝ ( ) ( )⎠ Proof. First, check that M SL2 R φ M SL2 S . Indeed, if M

SL2 R , then det M ad ∈ bc (1R). NowÔ⇒ note( that) ∈ det (φ M) ψ a ψ d ∈ ψ b (ψ )c . Using all( three) = properties− = from definition 30, this( ( can)) be= rewritten( ) ( ) as−

ψ(ad) ( )ψ bc then ψ ad bc ψ 1R 1S. Therefore, φ M is an element of

SL( 2 )S−. A( similar) process( − follows) = ( from) = the fact that matrix( multiplication) and addition( ) are preserved by the homomorphism.

24 The ability to induce a group homomorphism given a ring homomorphism makes declaring certain homomorphisms much easier later on, and allows several computations to be skipped. While lemma 4 shows that SL2 R is functional with respect to ring homomorphisms, the next lemma describes( its) relationship with direct sums of rings.

Lemma 5. The Special Linear Group over a direct product of rings splits into the direct product of Special Linear Groups over the direct summands, i.e.

SL2 R S SL2 R SL2 S

( ⊕ ) ≃ ( ) × ( ) Proof. This isomorphism works by natural construction. Each coefficient of ma- trices from SL2 R S is an , so the isomorphism works by splitting these ordered pairs( ⊕ into) their own matrices. Let ri R and si S. Then the isomorphism φ SL2 R S SL2 R SL2 S works∈ by the following∈ mapping ∶ ( ⊕ ) → ( ) × ( ) r1, s1 r2, s2 r1 r2 s1 s2 , ⎛(r3, s3)(r4, s4)⎞ ⎛⎛r3 r4⎞ ⎛s3 s4⎞⎞ ⎜ ⎟ ⎜⎜ ⎟ ⎜ ⎟⎟ ⎜ ⎟ ↦ ⎜⎜ ⎟ ⎜ ⎟⎟ ⎝( )( )⎠ ⎝⎝ ⎠ ⎝ ⎠⎠ Elements of the different rings do not mix in a direct sum (or product), so if the determinant of the left side is 1R, 1S , then the determinants on the right side will still be 1R and 1S. ( )

25 Lemma 5 is what makes all of this work relevant, because it justifies the state- ment in the introduction that finite quotients of the modular group, SL2 Z NZ can be decomposed into it parts: ( ~ )

r mi N pi i=1 r = M mi SL2 Z NZ SL2 Z pi Z i=1 ( ~ ) ≃  ( ~ ) 3 Conjugacy Classes of Matrix Groups

In this chapter, the symbol Fp will be used in place of Z pZ to emphasize the fact that we are working over a field. The goal of this chapter~ is to find and present conjugacy classes of SL2 Fp . First, the conjugacy classes in the larger group, GL2 Fp , will be determined( using) a number of algebraic tools. The natural place to begin( ) this investigation is with the order of the two groups in question.

After establishing their orders, conjugacy classes in GL2 Fp will be computed.

Finally, conjugacy classes in SL2 Fp will be computed and( certain) nuances (such as splitting of conjugacy classes)( will) be identified and discussed.

3.1 Order of the Matrix Groups

Definition 32. The of size two is the group of two by two nonsingular matrices with entries from a given ring.

a b GL2 Fp a, b, c, d R; ad bc 0 ⎧ R ⎫ ⎪ ⎛c d⎞ R ⎪ ⎪ ⎜ ⎟ R ⎪ ( ) = ⎨ ⎜ ⎟ R ∈ − ≠ ⎬ ⎪ R ⎪ ⎩⎪ ⎝ ⎠ R ⎭⎪

26 2 2 Proposition 3. The order of GL2 Fp is p 1 p p .

( ) ( − )( − ) Proof. Approach the construction of the matrix one column at a time. The first

2 column can be any pair a, b Fp except a b 0. This leaves p 1 options. By theorem 4, a matrix has non-zero∈ determinant= if= and only if its columns− are linearly independent. This implies the only restriction on the second column is that it is not a multiple of the first column. Given any (non-zero) vector, there are exactly p multiples, so the second vector is chosen from p2 p options. Therefore, we obtain the result − 2 2 GL2 Fp p 1 p p

S ( )S = ( − )( − )

From here, it is a relatively common exercise in an undergraduate abstract algebra course to calculate the order of SL2 Fp . This calculation begins by noting that the defined by ”taking the determinant”( ) can be viewed as a surjective homomorphism

∗ det GL2 Fp Fp

∶ ( ) → By the definition of GL2 Fp , this homomorphism is clearly defined for every el- ement in GL2 Fp , and lemma( ) 2 establishes that the multiplicative property of a 0 homomorphisms( holds.) To see surjectivity, note that the determinant of 0 1 is a, so every determinant value will be taken. ( )

3 Proposition 4. The order of SL2 Fp is p p.

( ) − Proof. Because SL2 Fp consists of matrices in GL2 Fp with determinant equal to one, it is clear that( the) kernel of the homomorphism( det) is exactly SL2 Fp . By ( )

27 theorems 2 and 1, this leads to the conclusion

GL p p 1 p2 1 SL 2 Fp p3 p 2 Fp ∗ Fp p 1 S ( )S ( − )( − ) S ( )S = = = − S S −

3.2 Conjugacy Classes of GL2 Fp

In order to determine conjugacy classes in( this) group, certain results from linear algebra will help greatly. Namely, the fact that the minimal polynomial of a matrix is invariant under conjugation provides an excellent place to start defining the classes. By considering the minimal polynomial associated with each conjugacy class, a primitive classification system can be constructed (and then refined). In a 2 2 matrix, the maximum degree of a minimal polynomial is 2. There- fore, all possible× minimal polynomials can be identified quickly. Every minimal polynomial will have precisely one of the following forms, where a, b, c, d Fp, such that a b, and c, d create an irreducible polynomial: ∈ ≠

Linear (one root) t a

Quadratic (two identical roots) t (a− t ) a

Quadratic (two distinct roots) (t − a)(t − b)

Quadratic (irreducible) ( t−2 )(ct −d )

− −

28 3.2.1 Linear Case

A matrix with a linear minimal polynomial (t a) must be be scalar, which is the simplest case. Scalar matrices commute with every− matrix (proposition 2), so the equation M N N M holds true for every scalar matrix M and every matrix

N that come∗ from= GL∗2 Z pZ . Commutative elements of a group are in their own (size 1) conjugacy classes,( ~ so) scalar matrices contribute p 1 individual conjugacy classes. −

3.2.2 First Quadratic Case

When the minimal polynomial of a matrix is a perfect square (of the form t a 2),

the matrix is conjugate to the matrix ( − )

a 1 ∗ ; a Fp (1) ⎛0 a⎞ ⎜ ⎟ ⎜ ⎟ ∈ ⎝ ⎠ Proposition 5. The centralizer of matrix (1) has p2 p elements and consists of

matrices of the form −

z x ∗ ; z Fp, x Fp ⎛0 z⎞ ⎜ ⎟ ⎜ ⎟ ∈ ∈ ⎝ ⎠ Proof. First, the centralizer is computed explicitly. If matrix (1) is denoted A, and

29 a matrix with free coefficients w, x, y, and z is denoted P .

a 1 w x aw y ax z AP ⎛0 a⎞ ⎛y z⎞ ⎛ ay+ az+ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ ⎝w x⎠ ⎝ a 1⎠ ⎝aw ax w ⎠ PA ⎛y z⎞ ⎛0 a⎞ ⎛ay az + y ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ + ⎠ By equating these matrices term for term, the top left entries reveal y 0 and the top right entries yield w z. Finally, to keep a non-zero determinant, z=cannot be zero. This returns the original= result from the proposition.

∗ Second, note that x Fp gives p options for x, and z Fp gives p 1 choices for z. This gives a total∈ of p p 1 matrices, fulfilling∈ the second part− of the proposition. ( − )

With the size of the centralizer computed, the counting formula (theorem 3) can be used to find the size of these conjugacy classes, putting exactly

p p 1 p2 1 p2 1 p p 1 ( − )( − ) = − ( − ) elements in each of the p 1 conjugacy classes.

− 3.2.3 Second Quadratic Case

If the minimal polynomial of a matrix is t a t b , a b, then the matrix will be conjugate to the ( − )( − ) ≠

a 0 ∗ ; a, b Z pZ , a b ⎛0 b⎞ ⎜ ⎟ ⎜ ⎟ ∈ ( ~ ) ≠ ⎝ ⎠

30 If the above matrix is labeled A, then similarly to the derivation the first quadratic case, the centralizer can be calculated explicitly by multiplication of a generic matrix, P.

w x a 0 aw bx PA ⎛y z⎞ ⎛0 b⎞ ⎛ay bz⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ ⎝a 0 ⎠ ⎝w x⎠ ⎝aw ax⎠ AP ⎛0 b⎞ ⎛y z⎞ ⎛ yb bz⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ Because a and b were selected to be distinct, the only way ay by and ax bx is if x y 0. This proves that the centralizer of these matrices= will consist= of precisely= all= diagonal matrices, of which there are p 1 2. This puts ( − ) p p 1 p2 1 p p 1 (2) p 1 2 ( − )( − ) = ( + ) ( − ) p−1 (p−1)(p−2) elements in each of these conjugacy classes. With 2 2 ways to choose 1 distinct values for a and b, this case contributes‰ exactlyŽ = 2 p p 1 p 1 p 2 elements to the group. ( + )( − )( − )

3.2.4 Third Quadratic Case

In the third quadratic case, the minimal polynomial of a matrix has the form

2 ∗ t at b, where a Fp and b Fp such that the minimal polynomial is irreducible over− F−p. The total∈ number of∈ ( conjugacy) classes in this case is equal to the number of quadratic irreducible polynomials, which in turn is also equal to the total number of polynomials minus those that are reducible. To start, note that the total number of monic quadratic polynomials is p2 (by independently choosing a and b). Reducible polynomials can always be written

31 p as t a t b , so there are 2 ways to create these polynomials with distinct a

and( b−. Finally,)( − ) there will be p‰moreŽ polynomials formed when a b. In total, this yields =

p p2 p p p 1 p2 p p2 p 2 2 2 − ( − ) − ‹  − = ( − ) − = which is the number of conjugacy classes in the third quadratic case. In order to compute centralizers of this case, it will help to have a matrix representative for those conjugacy classes. Recall from the discussion of canonical forms in section 2.2 that when the minimal polynomial is an irreducible of the form t2 at b, the matrix is conjugate to − − 0 b ⎛1 a⎞ ⎜ ⎟ ⎜ ⎟ which will serve as a representative⎝ of its⎠ own conjugacy class in SL2 Fp . To compute its centralizer, multiply on either side by a generic matrix to( get) the following calculations:

w x 0 b x wb xa

⎛y z⎞ ⎛1 a⎞ ⎛z yb +za ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ ⎝0 b ⎠ ⎝w x⎠ ⎝ yb+ zb⎠ ⎛1 a⎞ ⎛y z⎞ ⎛w ya x za⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ + + ⎠ From these calculations, the centralizer can be visualized as

0 b w yb CGL2( p) w, y Fp, w, y 0, 0 F ⎧ R ⎫ ⎛1 a⎞ ⎪ ⎛y w ya⎞ R ⎪ ⎜ ⎟ ⎪ ⎜ ⎟ R ⎪ ⎜ ⎟ = ⎨ ⎜ ⎟ R ∈ ( ) ≠ ( ) ⎬ ⎪ R ⎪ ⎝ ⎠ ⎪ ⎝ + ⎠ R ⎪ The determinant of these centralizing⎩ matricesR is w2 wya y2b. If both⎭ w and y

32 + − are zero, then this determinant will be zero, making an invalid matrix for GL2 Fp .

If only one of w and y is zero, then the determinant will be w2 or y2, respectively.( ) This leaves the case of neither being zero, where we rewrite the determinant as

2 w 2 w y y y a b . Note that this is simply the minimal polynomial of the matrix in −w question,(( ) + evaluated− ) at t y . This equation cannot be zero because the minimal polynomial cannot have= roots in the field (from irreducibility). Therefore, the determinant will never be zero with non-zero selections of w and y, leaving p2 1 elements in the centralizer. The class equation then yields a total of p p −1 elements in each conjugacy class. ( − )

3.2.5 Classes in SL2 Fp

( ) Now that the conjugacy classes of GL2 Fp have been defined, it is important to note specfically which classes lie inside( of SL) 2 Fp . By the definition of SL2 Fp , only matrices with determinant one will be in( this) group. If representatives( of) classes in GL2 Fp are in SL2 Fp , then the rest of that conjugacy class must also be in SL2 Fp (, so) only one representative( ) from each class needs to be considered.

a 0 ∗ In the( linear) case, representatives have the form 0 a , where a Fp. In a finite field, the only elements that square to one are 1 and ( 1, so) the only∈ scalar matrices 1 0 with determinant one are the identity and negative− identity matrices, 0 1 and

−1 0 0 −1 , which are each their own conjugacy classes. The first quadratic( case) is 1 1 (similar,) only being able to contribute the classes represented by the matrices 0 1

−1 1 and 0 −1 . ( ) The( second) quadratic case has a conjugacy class for every distinct, unordered

pair, a, b , and conjugacy classes will be in SL2 Z pZ when a b 1. The identity

and negative{ } identity have already been accounted( ~ for,) and neither⋅ = a nor b can be equal to zero, leaving p 3 2 conjugacy classes represented by matrices of the

( − )~ 33 a 0 form 0 a−1 .

Recall‰ fromŽ the third quadratic case of GL2 Fp that matrices have minimal

polynomial t2 at b and representatives of conjugacy( ) classes have the form M

0 b 1 a . The determinant− − of these matrices is 0 a b b. This gives a concise and= concrete( ) condition for membership in SL2 Fp ⋅ − = − ( ) M SL2 Fp b 1

∈ ( ) ⇐⇒ = − The minimal polynomial of this third quadratic case now takes the form t2 at 1

where a is chosen so that the polynomial is irreducible. Intuitively one might− think+

there are p options for a Fp, but some of these values will result in reducible

polynomials. Fortunately,∈ these reducible polynomials are easy to compute. If the polynomial t2 at 1 is reducible, then it can be written as t c t d ,

c, d Fp. However, in− order+ for the constant term to be 1, d must( − equal)( − the) multiplicative∈ inverse of c, creating the minimal polynomial t c t c−1 t2

−1 ∗ c c t 1. Note that c must be invertible, i.e. c Fp. Additionally,( − )( − the) = cases− of( +c 1) and+ c 1 must be counted separately, because∈ those are the only cases where= c c−1.= This− leaves p 3 options when picking c, only half of which are valid because= picking c−1 will− duplicate the case of picking c. Note that for the

polynomial to be quadratic, a only needs to be selected from Fp, leaving p options,

p−3 p−1 2 so there are p 2 2 2 values of a that make t at 1 irreducible. This p−1 translates to having− − 2 distinct= conjugacy classes from the− third+ quadratic case of

GL2 Fp falling inside of SL2 Fp .

( ) ( )

34 3.3 Splitting of Classes of SL2 Fp

Now that conjugacy classes in GL2 Fp have( been) classified ( a representa- tive), classes in SL2 Fp need to be( considered.) There is a lemma from abstract algebra that will be( imperative) to this step.

Lemma 6. Given H G, and x H, the G conjugacy class of x is a of conjugacy classes in H◁. More specifically,∈ a− exists between the number of H conjugacy classes in the G conjugacy class of x and the double-coset space

G H − CG(x). −

Proof.Ó ÒTake x H. Note that xG H. Conjugate x with each of g, f G. By definition of the∈ equivalence relation⊂ imposed by conjugation, if gxg−1 is conjugate∈ to fxf −1 then there exists some h H such that

∈ hfxfh−1 gxg−1

g−1hf x = x g−1hf

−1 Ô⇒ ( g hf) = C(G x )

Ô⇒ hf ∈ gCG( x)

−1 Ô⇒ f ∈ h gC( G) x

Ô⇒ ∈ ( )

In the last line, h−1 can be exchanged for H. This means that gxg−1 will be only

−1 be conjugate to fxf if f is an element of the HgCG x . Directly following is the observation that choosing g to be from different cosets( creates) new sets of conjugates, so xG gxg−1 H g∈ HƒG~CG(x) = ( )

35 In our case, H SL2 Fp and G GL2 Fp . Lemma 6 will allow for a relatively

straightforward calculation= ( ) to determine= ( how) conjugacy classes split. In section

3.1, it was noted that SL2 Fp is the kernel of the determinant map from GL2 Fp

into Fp, so by Theorem (1)( it) is clear that ( )

GL2(Fp) ∗ SL2(Fp) Fp

Ó ≃ Because of the equivalence relation established by this factor group, finding the size

GL ( ) of the double coset space SL 2 Fp C h will be the same as finding 2(Fp) GL2(Fp)( ) ∗ Fp cosets of D where D det gÓ g CGLÒ2(Fp) h . There areÒ relatively few= ™ conjugacy( ) T ∈ class representatives( ) ž that need to run this check, and much of the relevant information has already been calculated in previous

sections. To start, consider the identity matrix. The GL2 Fp -centralizer of the

∗ identity is obviously GL2 Fp , whose determinants span all( of)Fp. Plugging into the double coset formula yields( ) the quotient group

∗ Fp ∗ 1 (3) Fp Ò ≃ { } This means that the GL2 Fp -conjugacy class of the identity matrix does not split

when passing into SL2 Fp( , which) is what we should expect because this class only has one element. An analogous( ) argument follows for the negative identity case. 1 1 The first non-trivial case is the GL2 Fp -conjugacy class of 0 1 . Recall from

section 3.2.2 that the GL2 Fp -centralizer( of) this conjugacy class( is) ( ) x y ∗ x Fp, y Fp ⎧ R ⎫ ⎪ ⎛0 x⎞ R ⎪ ⎪ ⎜ ⎟ R ⎪ ⎨ ⎜ ⎟ R ∈ ∈ ⎬ ⎪ R ⎪ ⎩⎪ ⎝ ⎠ R ⎭⎪

36 The determinant of any matrix in this group is x2. This means that the deter-

∗ ∗ 2 minants of matrices in this group will always be in Fp, denoted Fp . In

∗ 2 order to visualize the double-coset space, we need to know the size of F(p .)

∗ 2 p−1 ( ) Proposition 6. The order of Fp is 2 .

( ) ∗ ∗ Proof. First, note that the order of Fp is p 1. Given any element a Fp, we have

2 2 ∗ a a . This means that only half of− the elements of Fp are squares,∈ which

∗ 2 p−1 gives= ( the− ) result Fp 2 .

S( )∗ S2= ∗ The index of Fp in Fp is 2, leading to the conclusion that this GL2 Fp -

conjugacy class will( split) into two SL2 Fp -conjugacy classes. To find a represen-( )

tative of this second class, a conjugation( that) exists in GL2 Fp but is unattainable

∗ in SL2 Fp needs to be found. To do this, recall that Fp is a( cyclic) group which has

∗ a generator.( ) Let ω be a generator of Fp, and consider the following conjugation

ω 0 1 1 ω−1 0 ω ω ω−1 0 1 ω (4) ⎛0 1⎞ ⎛0 1⎞ ⎛ 0 1⎞ ⎛0 1⎞ ⎛ 0 1⎞ ⎛0 1⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ 1 1⎠ This shows that the matrix in equation (4) is conjugate to 0 1 in GL2 Fp ,

but because the conjugating matrix has determinant ω 1, the( ) conjugation( ) is

−1 1 unattainable in SL2 Fp . An analogous case can be followed≠ for 0 −1 , and together these cases yield( ) the following two results ( )

GL2(Fp) SL2(Fp) SL2(Fp) 1 1 1 1 1 ω ⎛0 1⎞ ⎛0 1⎞ ⎛0 1⎞ ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ GL2(Fp)⎜ ⎟ SL2(F⎜p) ⎟ SL2(Fp) ⎝ 1⎠ 1 ⎝ 1⎠ 1 ⎝ ⎠ 1 ω

⎛−0 1⎞ ⎛−0 1⎞ ⎛−0 1⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ ⎝ − ⎠ ⎝ − ⎠ ⎝ − ⎠ 37 Next is the second quadratic case, where the roots of the minimal polynomial are multiplicative inverses. These are still diagonal matrices, which means the earlier calculation of the centralizer still holds true. The GL2 Fp -centralizer is the set of all diagonal matrices. The freedom of the determinant( ) in this set can easily be seen by setting the first value in the matrix to 1 and running through

∗ the other elements of Fp. As with the identity, this conjugacy class will not split when passed to SL2 Fp .

Finally, there is( the) third quadratic case. As before, consider the centralizer computed in the previous section. In SL2 Fp , b must be replaced with 1 to satisfy the determinant condition, so representatives( ) of classes and their centralizers− take the following form, where a still satisfies the quadratic irreducibility condition from the previous section.

0 1 w y CGL2( p) w, y Fp, w, y 0, 0 F ⎪⎧ R ⎪⎫ ⎛1 −a ⎞ ⎪ ⎛y w− ya⎞ R ⎪ ⎜ ⎟ ⎪ ⎜ ⎟ R ⎪ ⎜ ⎟ = ⎨ ⎜ ⎟ R ∈ ( ) ≠ ( ) ⎬ ⎪ R ⎪ ⎝ ⎠ ⎪ ⎝ + ⎠ R ⎪ ⎩⎪ 2 R2 ∗ ⎭⎪ Proposition 7. The polynomial w wya y takes all values of Fp.

+ + Proof. First, note that if y 0, then the polynomial will be w2. Now, assume y 0

2 w 2 w 2 and consider instead the polynomial= y y y a 1 . Because y is a square,≠

w 2 w we only need to consider whether or not(( )y + y+a )1 is always a square. In w order to see this, note that we can make a(( substitution) + + of) y τ, making this a

2 polynomial F τ τ aτ 1, τ Fp. Now it is a question− of how= many different values F τ can( ) take= in− Fp+, or the∈ order of the set ( ) F τ τ Fp

{ ( )S ∈ }

38 Consider the equation F t F s , which breaks down as follows:

( ) = ( ) t2 at 1 s2 as 1

−t2 +s2 = a t− s +

t s t s −a = 0( − )

( − )( + − ) = Therefore, F t F s whenever t s or t a s. This means that for every value t Fp, there( are) = at( most) two values= equal= to −F t , namely F t itself as well as F∈a t . To avoid double-counting any values, note( ) that ( ) ( − ) a t a t 2t a t 2 = − ⇐⇒ = ⇐⇒ = a −1 which means that for t 2 , the pre-image F t is only one element. This leaves p−1 a 2 values that are taken= twice, in addition to(F) 2 , wrapping back to the result ( ) p 1 F τ τ 1 Fp 2 − S{ ( )S ∈ }S = + ∗ 2 p−1 The order of Fp is only 2 , so F τ must take a non-square value. When

2 multiplied by y( , this) will give the rest( of) the non-square values in Fp. Therefore,

2 ∗ the polynomial w wya 1 attains every value in Fp.

This proposition+ allows+ for the final realization of this section, that the con- jugacy classes represented by irreducible quadratic polynomials do not split in

SL2 Fp .

( )

39 Minimal Polynomial Matrix Representative Size 1 0 t 1 1 0 1 1 0 (t − 1) Œ ‘ 1 0 1 −ω 0 t (c +t )c−1 Œ ‘ p p 1 0 ω−−1 1 1 ( − t)( 1−2 ) Œ ‘ 1 (p2+ 1) 0 1 2 1 ω ( − ) Œ ‘ 1 (p2 − 1) 0 1 2 1 1 t 1 2 Œ ‘ 1 (p2 − 1) 0 1 2 −1 ω ( + ) Œ ‘ 1 (p2 − 1) 0 −1 2 −0 1 t2 at 1 Œ ‘ p( p −1 ) 1 a− − − + Œ ‘ ( − ) Figure 4: Compiled results for conjugacy classes of SL2 Fp

( ) 3.4 Results of SL2 Fp

In this final section of chapter( ) 3, results are reviewed and presented in a more accessible manner, with a couple final checks to show everything has worked out

∗ correctly to help convince the reader nothing has been missed. In figure 4 c Fp, ω

∗ 2 is a generator of the Fp, and a is selected so that t at∈ 1 is irreducible over Fp. Note that choosing a different ω will not change the conjugacy− + p−3 class. There are 2 choices for c, and in section 3.2.4 it was shown that there p−1 are 2 choices for a. The number of conjugacy classes in SL2 Fp can easily be computed as ( ) p 3 p 1 6 p 4 2 2 − − + + = + because of the options for a and c, as well as the other six conjugacy classes from the table. Further, the elements that have been accounted for can be totaled as

40 follows

p 3 p 1 1 Accounted elements 1 1 p p 1 p p 1 4 p2 1 2 2 2 p 3 − p 1 − { } = 2 + + p2 (p + ) + p2 (p − 2)p+2 2( − ) 2 2 −p3 p2 3p2 3−p p3 p2 p2 p = 2p+2 ( + ) + ( − ) + − 2 2p3+ 4p−2 2p− + − − + = 2p2 + 2 − − = p3 +p

= − 3 This matches the result from proposition 4 that the order of SL2 Fp is p p, which means all elements have been accounted for. ( ) −

2 4 Toward Lifting to SL2 Z p Z

The calculations of the previous chapter were( ~ very) convenient because they hap- pened to be over a field. However, one goal of this paper is to extend these results not to larger fields, but instead over the ring Z p2Z. This ring is not a field, making many abstract computations more difficult. To~ accomplish this, a surjective homo-

2 morphism is constructed from SL2 Z p Z to SL2 Z pZ to see what conjugacy

2 classes look like in SL2 Z p Z . Unfortunately,( ~ ) the( results~ ) of chapter 4 do not yet allow conjugacy classes( to~ be) explicitly computed, so it is harder to visualize the results in a form such as figure 4. Additionally, the use of Fp from here on will be suspended to emphasize the relationship between the base rings under SL2 R .

( )

41 4.1 Reduction mod p

Consider the ring homomorphism

2 modp Z p Z Z pZ (5)

∶ ~ → ~ defined by reducing elements of Z p2Z p.

~ Proposition 8. The kernel of modp is isormorphic to Z pZ.

~ Proof. This proof proceeds by simply asking the question ”What elements of Z p2Z

are congruent to 0 mod p?” By definition, these will be numbers that are divisible~ by p, which in turn are numbers writable as ap for some a Z p2Z. Now note that

these elements will only be distinct up to a being in Z pZ∈, because~ if a p then the a p b for some b Z pZ and the p2 term gets reduced~ to 0. This kernel≥ must be a group= + because it∈ is a~ normal subgroup of Z p2Z, and because it has prime order it must be isomorphic to Z pZ. ~

The check that equation 5 is a~ well-defined homomorphism is omitted because

modulo is well-known to pass through multiplication and addition. Because modp is a ring homomorphism, lemma 4 can be applied to induce a group homomorphism

2 φ SL2 Z p Z SL2 Z pZ (6)

∶ ( ~ ) → ( ~ ) Proposition 9. The map φ as defined above is a surjective homomorphism.

Proof. Because φ was induced by modp, lemma 4 tells us that this homomorphism is well-defined. In order to show that φ is surjective, recall a result from [Serre,

1 1 1973], which says that SL2 Z is generated by two matrices, S 0 1 and T

0 1 −1 0 . Because of this fact,( showing) that φ is surjective is equivalent= ( to) showing S=

( ) 42 2 and T have preimages in SL2 Z p Z . The last step is to recognize that φ S S and φ T T . The fact that SL( ~2 Z )pZ is generated by S and T means that( ) any= element( ) can= be written as the multiplication( ~ ) of a of S’s and T ’s, which can again be rewritten as compositions of φ S and φ T . These can then be collected under one operation of φ using the multiplicative( ) ( ) property from definition 10.

2 Therefore, every element of SL2 Z pZ has a pre-image in SL2 Z p Z , so φ is surjective. ( ~ ) ( ~ )

4.2 Exploring the Kernel

Now that φ has been shown to be a surjective homomorphism, lemma 2 can be used 2 SL2 Z p Z to make an isomorphism between ker φ and SL2 Z pZ . Recalling definition 14, the kernel of a homomorphism( ~ is) the set of elements that get mapped Ò ( ) ( ~ ) to the identity. This leads to the following proposition:

Proposition 10. For φ as described in equation 6,

a b ker φ I2 p a, b, c, d Z pZ, a d 0 ⎧ R ⎫ ⎪ ⎛c d⎞ R ⎪ ⎪ ⎜ ⎟ R ⎪ ( ) = ⎨ + ⎜ ⎟ R ∈ ~ + = ⎬ ⎪ R ⎪ ⎪ ⎝ ⎠ R ⎪ Proof. First, note that ker⎩ φ will consistR of those special linear matrices⎭ that map to the identity matrix. These( ) matrices can be identified by looking at the preimage of the identity matrix, which from lemma 4 we know can be examined component- wise. Under the ring homomorphism modp, the preimage of 1 is elements of the form px 1 and the preimage of 0 is elements of the form px. This means elements of the kernel+ take the form

1 0 pa 1 pb a b −1 modp I2 p ⎛0 1⎞ ⎛ pc+ pd 1⎞ ⎛c d⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ = + ⎜ ⎟ ⎝ ⎠ ⎝ 43 + ⎠ ⎝ ⎠ where a, b, c, and d come from ker modp , which by proposition 8 we know is

isomorphic to Z pZ. ( ) To see condition~ that a d 0, simply compute the determinant of the matrices in the kernel. The calculation+ = proceeds as follows

pa 1 pb det pa 1 pd 1 p2bc ⎛ pc+ pd 1⎞ ⎜ ⎟ ⎜ ⎟ = ( + )( + ) − ⎝ + ⎠ p2ad p a d 1

= p a +d ( 1+ ) +

= ( + ) + Multiples of p2 get reduced to zero, leaving just p a d 1, which means the

matrix will be in SL2 Z pZ if and only if a d is( congruent+ ) + to 0 mod p. This gives final conditions on( ~ the) kernel as +

a b ker φ I2 p a, b, c, d Z pZ, a d 0 ⎧ R ⎫ ⎪ ⎛c d⎞ R ⎪ ⎪ ⎜ ⎟ R ⎪ ( ) = ⎨ + ⎜ ⎟ R ∈ ~ + = ⎬ ⎪ R ⎪ ⎩⎪ ⎝ ⎠ R ⎭⎪

The condition that a d 0 is going to be referred to as the matrix being

traceless. + =

Proposition 11. The kernel of the group homomorphism φ is an abelian group under multiplication, and is isomorphic to the of 2 2 traceless

matrices. ×

Proof. Let M and N be elements of ker φ . Then M and N respectively have forms I2 pX and I2 pY where X and Y( are) 2 2 matrices with zero trace and + + ×

44 coefficients from Z pZ. Then MN and NM can be computed as follows:

~ 2 MN I2 pX I2 pY I2 p X Y p XY I2 p Y X

= ( + )( + ) = + ( + ) + = + ( + ) Matrix multiplication in ker φ behaves like matrix addition. Now, an isomor- phism from ker φ to M2 Z(pZ) 0 can be established by sending any element of the form I2 pX( ) ker φ( to~ X) M2 Z pZ 0. The check that this preserves the properties( of+ an isomorphism) ∈ ( ) fall directly∈ ( ~ from) the above calculation of MN.

Proposition 12. The order of ker φ is p3.

( ) Proof. First, note from proposition 11 that we can instead compute the order of

M2 Z pZ 0. This matrix has four variables restricted by being traceless. Therefore, if a(is~ selected,) then d a. This leaves three free variables to be chosen in Z pZ, which gives an order of=p−3. ~

2 3 3 Proposition 13. The order of SL2 Z p Z is p p p .

( ~ ) ( − ) Proof. Apply theorem 2 and Lagrange’s Theorem to obtain the isomorphism and the implication

2 SL p2 SL2 Z p Z SL p 2 Z Z SL p ker φ 2 Z Z ker φ 2 Z Z ( ~ ) S ( ~ )S Ò ≃ ( ~ ) Ô⇒ 2 = S ( ~ )S ( ) SLS2 Z(p)SZ ker φ SL2 Z pZ

2 3 3 Ô⇒ SSL2(Z~p Z)S = Sp p( )Sp⋅ S ( ~ )S

Ô⇒ S ( ~ )S = ( − ) where the last line is justified by propositions 4 and 12.

45 i 4.3 Generalizing to SL2 Z p Z

The results from the last section( extend~ very) naturally over homomorphisms of the form

i i−1 φi SL2 Z p Z SL2 Z p Z

∶ ( ~ ) → ( ~ ) Knowing various properties of this sequence of homomorphisms will help im- mensely in deducing facts about their conjugacy classes. In fact, the hope is that this will eventually lead to a recursive process, if not a formula, for computing conjugacy classes for arbitrary powers of p.

First, note that the argument to show that φi is surjective extends seamlessly to

1 1 0 1 φi. The matrices 0 1 and −1 0 generate SL2 Z , so they will also be generators

i of SL2 Z p Z for( any) i (N, and) their preimages( ) will always exist in the next power of( p~. ) ∈

Next, ker φi will be found exactly the same way. The kernel of some φi will take the form( )

a b i ker φi I2 p a, b, c, d Z pZ; a d 0 ⎧ R ⎫ ⎪ ⎛c d⎞ R ⎪ ⎪ ⎜ ⎟ R ⎪ ( ) = ⎨ + ⎜ ⎟ R ∈ ~ + = ⎬ ⎪ R ⎪ ⎪ ⎝ ⎠ R ⎪ The only difference from⎩ the base case isR that pi is pulled out front⎭ instead of just p. The same trick is applied to this subgroup as was in equation ?? to simply write the coefficients as coming from Z pZ. Checks to confirm this kernel is contained

i within SL2 Z p Z and that it is~ abelian follow directly from section 4.2.

3 i The order( ~ of the) kernel is still obviously p , but now the order of SL2 Z p Z will have an extra nuance. Lagrange’s Theorem now needs to be applied recursively( ~ )

46 to obtain the final result of this study.

i+1 i SL2 Z p Z ker φi SL2 Z p Z

3 i S ( ~ )S = Sp (SL)S2 ⋅ZS p Z( ~ )S

3 3 i−1 = p ⋅ Sp SL( 2~ Z )Sp Z

3 3 = p ⋅ ...⋅ S p (SL~2 Z p)SZ

= p3i⋅ p3⋅ p⋅ S ( ~ )S

= ⋅ ( − ) This process just peels off a p3 for each power of Z pZ, then evaluates the order of the base group. ~ Recalling definition 16, the situation that has been described is that of an . If ker φi is denoted Ki, then most of chapter 4 is describing an exact sequence of the form( )

i+1 φi i 1 Ki SL2 Z p Z SL2 Z p Z 1

→ → ( ~ ) Ð→ ( ~ ) → [Tang and Tseng, 2013] specifically addresses the question of how many conjugacy

i classes are in the preimage of a given conjugacy class in SL2 Z p Z when Ki is abelian. Unfortunately, the statement and application of this( result~ ) is anything but trivial, and requires a longer and much more in-depth study.

47 5 Closing Remarks

5.1 Future Work

The obvious direction for this work to continue in is to design a systematic way to

i compute conjugacy classes for SL2 Z p Z . With what has been found so far, it

2 seems that once this has been done( for~SL2) Z p Z , the rest of the cases will likely follow quickly. Knowing the conjugacy classes( ~ of a) group can be useful for several reasons, but one of the biggest applications of this knowledge is . Knowing the conjugacy classes of these groups would allow one to construct linear representations of the group, enabling the use of powerful linear algebra results for further analysis of the group’s behavior. Representations have been

i defined for SL2 Z pZ , but not for SL2 Z p Z in general, making this the natural next step in the( research~ ) process. ( ~ )

5.2 Conclusion

This was only one way of many to approach the computation of conjugacy classes for SL2 Z pZ , but it sheds light on this group’s relationship with linear algebra.

Computing( ~ conjugacy) classes for any specific group (say, SL2 Z 7Z could easily be accomplished by using a computer to explicitly calculate every( ~ option,) but no structure is revealed in that scenario. By keeping the case generalized to being over Z pZ we are still able to keep the big picture in mind and see it is easier to visualize~ some of the structure of the conjugacy classes.

Beyond the base case of SL2 Z pZ , the preliminary results are promising.

The sequence of homomorphisms φ( i ~from) section 4.3 behave surprisingly well, one example being the fact that the kernel of φi is always abelian. These results should provide an excellent starting place for more exciting work in this field.

48 Acknowledgments

First and foremost, I would like to thank my advisor, Dr. Alexei Davydov, for his unwavering support throughout this project. He always knew when to give the right advice, and his willingness scrutinize every detail of this thesis were instrumental in its completion. His kindness and his care for his student is truly remarkable.

I would like to thank Dr. Todd Eisworth for his academic guidance as well as his extraordinary patience with my late scheduling and general disorganization.

I am extremely grateful to Shehzad Ahmed, Darren Simmons, Dan Bossaller, and all the other Ohio University mathematics graduate students that helped guide me through the thesis-writing process. They have listened to me talk about my the- sis more than anyone else, and on numerous occasions helped me past academic ob- stacles.

I would like to express my gratitude to Cary Frith for always taking my walk- in appointments. She was typically the first person to know when something was going horribly wrong, and 9 times out of 10 she was the person that fixed it.

Finally, I would like to thank my parents for listening to my endless explana- tions of my research and offering emotional support regardless.

49 References

Larry Dornhoff. Theory. Marcel Dekker, Inc., 1971.

David S. Dummit and Richard M. Foote. Abstract Algebra. John Wiley & Sons, Inc., 2004.

James E. Humphreys. Representations of sl 2, p . The American Mathematical

Monthly,, 82(1):21–39, Jan 1975. ( )

Victor Kostyuk. Flat life, 2009. URL http://www.math.cornell.edu/~mec/ Winter2009/Victor/part1.htm.

Yassine Mrabet. Simple torus, Nov 2007. URL https://commons.wikimedia. org/wiki/File:Simple_Torus.svg.

Mark Reeder. Characters of sl2 q . 2008.

( ) Jean-Pierre Serre. A Course in Arithmetic. Springer-Verlag, 1973.

Xiang Tang and Hsian-Hua Tseng. Conjugacy classes and characters for extensions of finite groups. 2013.

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