MATHIEU GROUPS. One construction of the Mathieu groups involves Steiner systems, so we will begin by discussing Steiner systems.

Save this PDF as:

Size: px
Start display at page:

Download "MATHIEU GROUPS. One construction of the Mathieu groups involves Steiner systems, so we will begin by discussing Steiner systems."

Transcription

1 MATHIEU GROUPS SIMON RUBINSTEIN-SALZEDO Abstract. In this essay, we will discuss some interesting properties of the Mathieu groups, which are a family of five sporadic finite simple groups, as well as some closely related topics. One construction of the Mathieu groups involves Steiner systems, so we will begin by discussing Steiner systems. 1. Steiner systems Definition 1. Let l < m < n be positive integers. We say that a collection S 1, S 2,..., S N of distinct subsets of {1, 2,..., n} is an (l, m, n)-steiner system (denoted S(l, m, n)) if it satisfies the following two properties: For each i, S i = m. For every subset T {1, 2,..., n} with T = l, there is exactly one i so that S i T. Of course, Steiner systems do not exist for most triples (l, m, n), but occasionally they do exist. It can be shown, for instance, that S(2, 3, n) exists if and only if n 1 (mod 6) or 3 (mod 6). In our discussion of Mathieu groups, we will be most interested in the Steiner systems S(4, 5, 11), S(5, 6, 12), S(3, 6, 22), S(4, 7, 23), and S(5, 8, 24). Let us now show that these Steiner systems do in fact exist. In fact, we really only need to construct the Steiner systems S(5, 6, 12) and S(5, 8, 24); the others will follow easily from the constructions of these. In order to construct the Steiner system S(5, 6, 12), we will consider the projective line P 1 (F 11 ) over the field with eleven elements. This consists of the elements 0, 1, 2,..., 10, together with an additional element, which we will call. Now consider the set of squares in F 11, which ( are ) S 1 = {0, 1, 3, 4, 5, 9}. Now suppose a b γ PSL 2 (F 11 ). If γ is represented by, then define γ(s 1 ) = c d { az + b cz + d z S 1 }, Date: October 18,

2 2 SIMON RUBINSTEIN-SALZEDO where we interpret az+b to be if cz + d = 0. Then {γ(s cz+d 1) : γ PSL 2 (F 11 )} is a Steiner system S(5, 6, 12). To construct S(4, 5, 11) from S(5, 6, 12), we simply take those elements of S(5, 6, 12) that do not contain. Thus we have constructed the first two of our five Steiner systems. We now construct the Steiner system S(5, 8, 24). Perhaps the simplest way of describing it is by using binary lexicodes, as follows: Let b 1 be the 24-bit binary string consisting of all 0 digits. Now, for 2 n 4096, define b n to be the first string lexicographically that differs from each of the strings b 1, b 2,..., b n 1 in at least 8 positions. Of these 4096 strings we have just written down, 759 of them contain exactly eight 1 digits. For each of these 759 strings, take a subset S i of {1, 2,..., 24} which includes digit d if and only if the d th digit of the corresponding binary string is a 1. The subsets we have chosen form a Steiner system S(5, 8, 24). To construct S(4, 7, 23) and S(3, 6, 22) from S(5, 8, 24), we consider those S i S(5, 8, 24) that do not contain 24 in the first case, and 23 or 24 in the second. It should be pointed out here that Steiner systems are very rare. In particular, there are no known Steiner systems S(l, m, n) with l > 5, and the only ones known with l = 5 are S(5, 6, 12) and S(5, 8, 24), described here. 2. Mathieu groups Now that we have defined all the Steiner systems we need, we are able to define the Mathieu groups. These will be the automorphism groups of the Steiner systems. Definition 2. The Mathieu groups M 11, M 12, M 22, M 23, and M 24 are defined as follows: M 11 = {σ S 11 : σ(s) S(4, 5, 11) for all S S(4, 5, 11)}. M 12 = {σ S 12 : σ(s) S(5, 6, 12) for all S S(5, 6, 12)}. M 22 = {σ S 22 : σ(s) S(3, 6, 22) for all S S(3, 6, 22)}. M 23 = {σ S 23 : σ(s) S(4, 7, 23) for all S S(4, 7, 23)}. M 24 = {σ S 24 : σ(s) S(5, 8, 24) for all S S(5, 8, 24)}. The Mathieu groups are the five smallest of the 26 sporadic finite simple groups. Definition 3. Let G be a group which acts on a set X. We say that G is k-transitive if for any two k-tuples (x 1,..., x k ) and (y 1,..., y k ) of elements of X, where all the x i s are distinct, and all the y i s are distinct, there is some g G so that g(x i ) = y i for 1 i k. It follows fairly easily from the definitions that M 22 is 3-transitive, M 11 and M 23 are 4-transitive, M 12 and M 24 are 5-transitive.

3 Theorem 4. The five Mathieu groups are simple. MATHIEU GROUPS 3 We ll prove this theorem in bits, beginning with M 11 and M 23, which Robin Chapman proves elegantly in [3]. We follow his treatment for the simplicity of M 11 and M 23, and we follow Alder in [1] for the proofs of the simplicity of M 12, M 22, and M 24. We begin with some basic facts about permutation groups. Let p be a prime and X = {1,..., p}. If G S p with G = n, then G acts on X transitively if and only if p n. Suppose now that G is transitive, so that p n, and let P be a Sylow p-subgroup of G. Let m G denote the number of Sylow p-subgroups of G so that m G = G : N G (P ), and let r G = N G (P ) : P. Thus n = pr G m G. Sylow s Theorem tells us that m G 1 (mod p). Now, since P N G (P ) N Sp (P ) and N Sp (P ) = A GL 1 (F p ), the affine general linear group of invertible affine maps x ax + b (mod p), we have p = P N G (P ) N Sp (P ) = p(p 1), so r G (p 1). Lemma 5. Suppose G is a transitive subgroup of S p, and r G = 1. Then G = Z/pZ. Proof. Suppose r G = 1. Then G contains m G (p 1) = n m G elements of order p. Since elements of order p do not fix any points of X, G contains at most m G elements which fix at least one point of X. Now, stabilizers G j of j X each contain G /p = m G elements, so all elements of G other than those of order p fix all points of X. But since G acts faithfully on X, it follows that m G = 1, so n = p, or G = Z/pZ. Lemma 5 already allows us to show that certain subgroups of S p are simple. Theorem 6. Let G S p act transitively on X, where G = pmr with m > 1, m 1 (mod p) and r < p prime. Then G is a simple group. Proof. Clearly, in this case, r G = r and m G = m. Suppose H is a nontrivial normal subgroup of G. Then the H-orbits of X are permuted transitively by G, so all the orbits have the same size. Thus X consists of a unique H-orbit, so H acts transitively on X. Thus p H, so H has some Sylow p-subgroup Q. Since H G, H contains all Sylow p-subgroups of G (since they are all conjugate), so m H = m G = m. Thus H = pmt for some t r. But t > 1 by Lemma 5, and r is prime by hypothesis, so H = G. Thus G is simple. In order to show that the Mathieu groups M 11 and M 23 are simple, we need to know something about their orders. Lemma 7. M 11 = M 23 = We can now put all this information together to show that M 11 and M 23 are simple. Theorem 8. M 11 and M 23 are simple.

4 4 SIMON RUBINSTEIN-SALZEDO Proof. We first check that M 11 is simple. Since 11 M 11, M 11 acts transitively on {1,..., 11}. Since M 11 /11 = (mod 11), we have r G = 5 and m G = 144. Thus Theorem 6 tells us that M 11 is simple. We now check that M 23 is simple. Now, 23 M 23, so M 23 acts transitively on {1,..., 23}. Since M 23 /23 = (mod 23) so r G = 11 and m G = Again Theorem 6 tells us that M 23 is simple. In order to prove the simplicity of M 12 and M 24 we cite (without proof) a lemma from Rotman [6], p Lemma 9. Suppose G acts faithfully and k-transitively on a set X. If there is some x X so that the stabilizer G x is simple, then If k 4, G is simple. If k 3 and X 2 r for any r, then either G = S 3 or G is simple. Together with Theorem 8, Lemma 9 immediately implies the following result: Theorem 10. The Mathieu groups M 12 and M 24 are simple. The one remaining Mathieu group is M 22. This again follows from Lemma 9 once we note that a point stabilizer of M 22 is PSL 3 (F 4 ), which is simple. Hence Theorem 11. M 22 is simple. 3. Binary Golay Codes Let us return to the code we used to construct the Steiner system S(5, 8, 24). Recall that this code consists of those 24-digit binary strings which are lexicographically first with respect to differing from all prior strings in at least eight positions. We call this code the extended Golay code. If we delete the rightmost digit in each codeword of the extended Golay code, we obtain the perfect Golay code. The perfect Golay code is very important in coding theory, for it allows us to send data which may be corrupted as efficiently as possible. More precisely, suppose we wish to send a string of twelve binary digits, but we know that up to three of the digits in any string we send may be changed. Then, by sending a certain 23-digit string with the given twelve initial digits, we can correct any three (or fewer) errors in order to recover the original string. Furthermore, with the Golay code, this recovery is optimal, in the following sense: let f : F 12 2 F 23 2 be the encoding of data (so that the first twelve digits of f(s) are equal to those of S). Then any element of F 23 2 differs from some element of f(f 12 2 ) in at most three digits, so any string in F 23 2 encodes a unique string from F 12. We mentioned earlier that M 24 is the automorphism group of the set of codewords with exactly eight 1 s. However, it is also true that M 24 is the automorphism group of the entire extended Golay code. Similarly, M 23 is the automorphism group of the perfect Golay code.

5 MATHIEU GROUPS 5 It turns out that the extended Golay code is linear, in the sense that if v, w F 24 2 are both in the Golay code, so is v+w. Thus the Golay code forms a 12-dimensional vector subspace of F To prove this, we need some results about impartial combinatorial games. We first introduce the game of nim. Proofs of all statements about general games can be found in either [2] or [4]. More details about the relation between codes and games can be found in [5]. In the game of nim, there are several piles of stones, and two players who will take turns making moves. A move consists of choosing one of the piles with at least one stone, and removing as many stones from that pile as are desired (but at least one). The winner is the player who removes the last stone. (Alternatively, the loser is the player whose turn it is to move when there are no more legal moves remaining.) We can generalize the game of nim so that we allow various different types of moves, but so that players still alternate moves and so that the unfortunate player who is to move but does not have any legal moves remaining is the loser. We will still have several piles of stones, but now we will have a certain family of so-called turning sets {h, i, j,...}, where h > i > j >, so that it is legal to replace a pile of size h with piles of size i, j,... A general position therefore is a sum of piles P a + P b + P c +, where P r denotes a pile of size r. Thus nim consists of turning sets {h, i} where h > i 0. We now define precisely what it means for certain positions to be winning for one player. Definition 12. Let G be an impartial combinatorial game of the type described above. Then we divide the class of positions of games into two outcome classes called N and P, in the following manner. We say that the zero position, denoted 0, consisting of no piles, is in class P. We then say: G N if there is some legal move G G so that G P, G P if all legal moves G G satisfy G N. The N positions (standing for N ext player) are those in which the player to move can win with optimal play, and the P positions (standing for Previous player) are those in which the player who just moved wins with optimal play. Since all are games are decided in a fixed number of moves, any G is either in class N or P. We now say what it means to add two games. If G and H are games, a move consists of making a legal move in exactly one of G and H. The game ends when there are no legal moves left in either game. Definition 13. We say two games G and H are equivalent (and write G = H) if whenever X is a game, either G + X P and H + X P, or G + X N and H + X N. That is, G and H act in the same way with respect to adding other games.

6 6 SIMON RUBINSTEIN-SALZEDO Theorem 14 (Sprague Grundy). Every impartial combinatorial game is equivalent to a unique nim game with exactly one pile. All P positions are equivalent to 0. A very important (but easy) corollary of the Sprague Grundy Theorem is that the sum of two P positions is again a P position. Much more generally, however, is the following: Let n denote the nim position with exactly one pile, consisting of n stones. Then m + n = (m n), where m n is the number obtained by writing m and n in binary and adding without carrying (alternatively, addition when considered as being elements of an F 2 -vector space). If P is some game position, write G(P ) = n if P = n. G(P ) is called the Grundy value of P. Since a game with two piles of the same size is clearly a P position, we do not change anything in the evaluation of a game by deleting piles of the same size in pairs. Thus without loss of generality, we may assume that for each n, the number of piles of size n is either 0 or 1. Therefore, we may express a game position in terms of a binary string a 3 a 2 a 1, where a i is the number of strings of size i (either 0 or 1). Since we can express any position as P = a 1 P 1 + a 2 P 2 + (where P i is a pile of size i), we have G(P ) = i 1a i G(P i ). Since the Grundy numbers of general positions are defined by a linear equation, we have the following result: Theorem 15. The set of binary strings corresponding to the P positions is F 2 -linear. We now consider the following game which helps us to prove the linearity of the Golay code. It suffices to describe the turning sets for the game; they are all sets of size one through seven so that the largest element is at most 24. (Since piles of size 25 or higher are irrelevant, we do not list them.) It is easy to check using Definition 12 that the words of the extended Golay code are exactly the P positions of this game. Hence Theorem 15 implies the following result: Theorem 16. The extended Golay code is F 2 -linear. 4. The Leech Lattice and other sporadic simple groups The Mathieu groups and the Golay code are related to a lattice in R 24, called the Leech lattice, which generates a sphere packing in 24 dimensions. The maximum number of spheres that can be tangent to a fixed sphere in R 24 so that all spheres have lattice point centers is 196,560. We now describe how this can be done by giving the coordinates of the centers. We take our fixed sphere to be centered at the origin in R 24. Of the 196,560 tangent spheres, we now describe the centers of 97,152 of them. If we fix a code work α with eight 1 s in the extended Golay code (alternatively, an element of the Steiner system S(5, 8, 24)), we generate 2 7 sphere centers from α by

7 MATHIEU GROUPS 7 placing ±2 in the positions in which α has a 1 in such a way that the number of 2 terms is even. We place a 0 everywhere else. This gives = 97, 152 points. We now describe another 1,104 sphere centers. We place ±4 in any two of the coordinates and 0 everywhere else. This gives 2 2 ( ) 24 2 = 1, 104 points. We now describe the remaining 97,308 sphere centers. One coordinate is ±3, and the rest are ±1. The sign choices are given by the words in the extended Golay code. We take the radius of each sphere to be 8. The Leech lattice allows us to construct more sporadic simple groups. For instance, the automorphism group of the Leech lattice contains the Conway group Co 1 as an index 2 subgroup, and the smaller Conway groups Co 2 and Co 3 are the stabilizers of one vector of the second and third type described above, respectively. The McLaughlin group, the Higman-Sims group, the Suzuki group, and the Hall- Janko group are also stabilizers of various sets of vectors in the Leech lattice. Thus straightfoward considerations of the Leech lattice provides us with seven of the sporadic groups, sometimes called the second generation of the Happy Family. (The five Mathieu groups are the first generation.) These sporadic groups related to the Leech lattice as well as the others are fascinating objects, but are unfortunately beyond the scope of this essay. References [1] Stuart Alder, Classical papers in group theory: the Mathieu groups, Master s thesis, University of East Anglia, [2] Elwyn R. Berlekamp, John H. Conway, and Richard K. Guy, Winning ways for your mathematical plays. Vol. 1, second ed., A K Peters Ltd., Natick, MA, MR MR (2001i:91001) [3] Robin J. Chapman, An elementary proof of the simplicity of the Mathieu groups M 11 and M 23, Amer. Math. Monthly 102 (1995), no. 6, MR MR [4] J. H. Conway, On numbers and games, second ed., A K Peters Ltd., Natick, MA, MR MR (2001j:00008) [5] John H. Conway and N. J. A. Sloane, Lexicographic codes: error-correcting codes from game theory, IEEE Trans. Inform. Theory 32 (1986), no. 3, MR MR (87f:94049) [6] Joseph J. Rotman, An introduction to the theory of groups, fourth ed., Graduate Texts in Mathematics, vol. 148, Springer-Verlag, New York, MR MR (95m:20001) Department of Mathematics, Stanford University, Stanford, CA address:

COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH. 1. Introduction

COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH ZACHARY ABEL 1. Introduction In this survey we discuss properties of the Higman-Sims graph, which has 100 vertices, 1100 edges, and is 22 regular. In fact

MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES

MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES 2016 47 4. Diophantine Equations A Diophantine Equation is simply an equation in one or more variables for which integer (or sometimes rational) solutions

NIM with Cash. Abstract. loses. This game has been well studied. For example, it is known that for NIM(1, 2, 3; n)

NIM with Cash William Gasarch Univ. of MD at College Park John Purtilo Univ. of MD at College Park Abstract NIM(a 1,..., a k ; n) is a -player game where initially there are n stones on the board and the

MATH10040 Chapter 2: Prime and relatively prime numbers

MATH10040 Chapter 2: Prime and relatively prime numbers Recall the basic definition: 1. Prime numbers Definition 1.1. Recall that a positive integer is said to be prime if it has precisely two positive

I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

Sowing Games JEFF ERICKSON

Games of No Chance MSRI Publications Volume 29, 1996 Sowing Games JEFF ERICKSON Abstract. At the Workshop, John Conway and Richard Guy proposed the class of sowing games, loosely based on the ancient African

LEARNING OBJECTIVES FOR THIS CHAPTER

CHAPTER 2 American mathematician Paul Halmos (1916 2006), who in 1942 published the first modern linear algebra book. The title of Halmos s book was the same as the title of this chapter. Finite-Dimensional

The determinant of a skew-symmetric matrix is a square. This can be seen in small cases by direct calculation: 0 a. 12 a. a 13 a 24 a 14 a 23 a 14

4 Symplectic groups In this and the next two sections, we begin the study of the groups preserving reflexive sesquilinear forms or quadratic forms. We begin with the symplectic groups, associated with

About the inverse football pool problem for 9 games 1

Seventh International Workshop on Optimal Codes and Related Topics September 6-1, 013, Albena, Bulgaria pp. 15-133 About the inverse football pool problem for 9 games 1 Emil Kolev Tsonka Baicheva Institute

160 CHAPTER 4. VECTOR SPACES

160 CHAPTER 4. VECTOR SPACES 4. Rank and Nullity In this section, we look at relationships between the row space, column space, null space of a matrix and its transpose. We will derive fundamental results

ON GALOIS REALIZATIONS OF THE 2-COVERABLE SYMMETRIC AND ALTERNATING GROUPS

ON GALOIS REALIZATIONS OF THE 2-COVERABLE SYMMETRIC AND ALTERNATING GROUPS DANIEL RABAYEV AND JACK SONN Abstract. Let f(x) be a monic polynomial in Z[x] with no rational roots but with roots in Q p for

Math 319 Problem Set #3 Solution 21 February 2002

Math 319 Problem Set #3 Solution 21 February 2002 1. ( 2.1, problem 15) Find integers a 1, a 2, a 3, a 4, a 5 such that every integer x satisfies at least one of the congruences x a 1 (mod 2), x a 2 (mod

Math 4310 Handout - Quotient Vector Spaces

Math 4310 Handout - Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable

Section 6.1 - Inner Products and Norms

Section 6.1 - Inner Products and Norms Definition. Let V be a vector space over F {R, C}. An inner product on V is a function that assigns, to every ordered pair of vectors x and y in V, a scalar in F,

Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

Nim Games. There are initially n stones on the table.

Nim Games 1 One Pile Nim Games Consider the following two-person game. There are initially n stones on the table. During a move a player can remove either one, two, or three stones. If a player cannot

Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,

Group Theory. Contents

Group Theory Contents Chapter 1: Review... 2 Chapter 2: Permutation Groups and Group Actions... 3 Orbits and Transitivity... 6 Specific Actions The Right regular and coset actions... 8 The Conjugation

Sudoku an alternative history

Sudoku an alternative history Peter J. Cameron p.j.cameron@qmul.ac.uk Talk to the Archimedeans, February 2007 Sudoku There s no mathematics involved. Use logic and reasoning to solve the puzzle. Instructions

GROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G.

Definition 1: GROUPS An operation on a set G is a function : G G G. Definition 2: A group is a set G which is equipped with an operation and a special element e G, called the identity, such that (i) the

15-251: Great Theoretical Ideas in Computer Science Anupam Gupta Notes on Combinatorial Games (draft!!) January 29, 2012

15-251: Great Theoretical Ideas in Computer Science Anupam Gupta Notes on Combinatorial Games (draft!!) January 29, 2012 1 A Take-Away Game Consider the following game: there are 21 chips on the table.

CS 3719 (Theory of Computation and Algorithms) Lecture 4

CS 3719 (Theory of Computation and Algorithms) Lecture 4 Antonina Kolokolova January 18, 2012 1 Undecidable languages 1.1 Church-Turing thesis Let s recap how it all started. In 1990, Hilbert stated a

Introduction to finite fields

Introduction to finite fields Topics in Finite Fields (Fall 2013) Rutgers University Swastik Kopparty Last modified: Monday 16 th September, 2013 Welcome to the course on finite fields! This is aimed at

Algebra of the 2x2x2 Rubik s Cube

Algebra of the 2x2x2 Rubik s Cube Under the direction of Dr. John S. Caughman William Brad Benjamin. Introduction As children, many of us spent countless hours playing with Rubiks Cube. At the time it

Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

NOTES ON LINEAR TRANSFORMATIONS

NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all

The Next Move. Some Theory and Practice with Impartial Games. Saturday Morning Math Group. March 1, 2014

The Next Move Some Theory and Practice with Impartial Games Saturday Morning Math Group March 1, 2014 Combinatorial games are two-player games with the following characteristics: Two players alternate

Similarity and Diagonalization. Similar Matrices

MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that

Codes for Network Switches

Codes for Network Switches Zhiying Wang, Omer Shaked, Yuval Cassuto, and Jehoshua Bruck Electrical Engineering Department, California Institute of Technology, Pasadena, CA 91125, USA Electrical Engineering

Stupid Divisibility Tricks

Stupid Divisibility Tricks 101 Ways to Stupefy Your Friends Appeared in Math Horizons November, 2006 Marc Renault Shippensburg University Mathematics Department 1871 Old Main Road Shippensburg, PA 17013

MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

THE DIMENSION OF A VECTOR SPACE

THE DIMENSION OF A VECTOR SPACE KEITH CONRAD This handout is a supplementary discussion leading up to the definition of dimension and some of its basic properties. Let V be a vector space over a field

Induction. Margaret M. Fleck. 10 October These notes cover mathematical induction and recursive definition

Induction Margaret M. Fleck 10 October 011 These notes cover mathematical induction and recursive definition 1 Introduction to induction At the start of the term, we saw the following formula for computing

Applications of Fermat s Little Theorem and Congruences

Applications of Fermat s Little Theorem and Congruences Definition: Let m be a positive integer. Then integers a and b are congruent modulo m, denoted by a b mod m, if m (a b). Example: 3 1 mod 2, 6 4

T ( a i x i ) = a i T (x i ).

Chapter 2 Defn 1. (p. 65) Let V and W be vector spaces (over F ). We call a function T : V W a linear transformation form V to W if, for all x, y V and c F, we have (a) T (x + y) = T (x) + T (y) and (b)

GROUPS ACTING ON A SET

GROUPS ACTING ON A SET MATH 435 SPRING 2012 NOTES FROM FEBRUARY 27TH, 2012 1. Left group actions Definition 1.1. Suppose that G is a group and S is a set. A left (group) action of G on S is a rule for

PYTHAGOREAN TRIPLES PETE L. CLARK

PYTHAGOREAN TRIPLES PETE L. CLARK 1. Parameterization of Pythagorean Triples 1.1. Introduction to Pythagorean triples. By a Pythagorean triple we mean an ordered triple (x, y, z) Z 3 such that x + y =

On an algorithm for classification of binary self-dual codes with minimum distance four

Thirteenth International Workshop on Algebraic and Combinatorial Coding Theory June 15-21, 2012, Pomorie, Bulgaria pp. 105 110 On an algorithm for classification of binary self-dual codes with minimum

Vieta s Formulas and the Identity Theorem

Vieta s Formulas and the Identity Theorem This worksheet will work through the material from our class on 3/21/2013 with some examples that should help you with the homework The topic of our discussion

Sec 4.1 Vector Spaces and Subspaces

Sec 4. Vector Spaces and Subspaces Motivation Let S be the set of all solutions to the differential equation y + y =. Let T be the set of all 2 3 matrices with real entries. These two sets share many common

Inner Product Spaces

Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and

= 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that

Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without

some algebra prelim solutions

some algebra prelim solutions David Morawski August 19, 2012 Problem (Spring 2008, #5). Show that f(x) = x p x + a is irreducible over F p whenever a F p is not zero. Proof. First, note that f(x) has no

G = G 0 > G 1 > > G k = {e}

Proposition 49. 1. A group G is nilpotent if and only if G appears as an element of its upper central series. 2. If G is nilpotent, then the upper central series and the lower central series have the same

Integer roots of quadratic and cubic polynomials with integer coefficients

Integer roots of quadratic and cubic polynomials with integer coefficients Konstantine Zelator Mathematics, Computer Science and Statistics 212 Ben Franklin Hall Bloomsburg University 400 East Second Street

Pigeonhole Principle Solutions

Pigeonhole Principle Solutions 1. Show that if we take n + 1 numbers from the set {1, 2,..., 2n}, then some pair of numbers will have no factors in common. Solution: Note that consecutive numbers (such

3. Mathematical Induction

3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)

1. Give the 16 bit signed (twos complement) representation of the following decimal numbers, and convert to hexadecimal:

Exercises 1 - number representations Questions 1. Give the 16 bit signed (twos complement) representation of the following decimal numbers, and convert to hexadecimal: (a) 3012 (b) - 435 2. For each of

11 Ideals. 11.1 Revisiting Z

11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(

Basics of Counting. The product rule. Product rule example. 22C:19, Chapter 6 Hantao Zhang. Sample question. Total is 18 * 325 = 5850

Basics of Counting 22C:19, Chapter 6 Hantao Zhang 1 The product rule Also called the multiplication rule If there are n 1 ways to do task 1, and n 2 ways to do task 2 Then there are n 1 n 2 ways to do

Linear Codes. Chapter 3. 3.1 Basics

Chapter 3 Linear Codes In order to define codes that we can encode and decode efficiently, we add more structure to the codespace. We shall be mainly interested in linear codes. A linear code of length

The Prime Numbers. Definition. A prime number is a positive integer with exactly two positive divisors.

The Prime Numbers Before starting our study of primes, we record the following important lemma. Recall that integers a, b are said to be relatively prime if gcd(a, b) = 1. Lemma (Euclid s Lemma). If gcd(a,

The Chinese Remainder Theorem

The Chinese Remainder Theorem Evan Chen evanchen@mit.edu February 3, 2015 The Chinese Remainder Theorem is a theorem only in that it is useful and requires proof. When you ask a capable 15-year-old why

CONTRIBUTIONS TO ZERO SUM PROBLEMS

CONTRIBUTIONS TO ZERO SUM PROBLEMS S. D. ADHIKARI, Y. G. CHEN, J. B. FRIEDLANDER, S. V. KONYAGIN AND F. PAPPALARDI Abstract. A prototype of zero sum theorems, the well known theorem of Erdős, Ginzburg

POWER SETS AND RELATIONS

POWER SETS AND RELATIONS L. MARIZZA A. BAILEY 1. The Power Set Now that we have defined sets as best we can, we can consider a sets of sets. If we were to assume nothing, except the existence of the empty

WHICH LINEAR-FRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE?

WHICH LINEAR-FRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE? JOEL H. SHAPIRO Abstract. These notes supplement the discussion of linear fractional mappings presented in a beginning graduate course

Testing Low-Degree Polynomials over

Testing Low-Degree Polynomials over Noga Alon, Tali Kaufman, Michael Krivelevich, Simon Litsyn, and Dana Ron Department of Mathematics, Tel Aviv University, Tel Aviv 69978, Israel and Institute for Advanced

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization

THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS

THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear

Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan

Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 3 Binary Operations We are used to addition and multiplication of real numbers. These operations combine two real numbers

by the matrix A results in a vector which is a reflection of the given

Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the y-axis We observe that

Matrix Norms. Tom Lyche. September 28, Centre of Mathematics for Applications, Department of Informatics, University of Oslo

Matrix Norms Tom Lyche Centre of Mathematics for Applications, Department of Informatics, University of Oslo September 28, 2009 Matrix Norms We consider matrix norms on (C m,n, C). All results holds for

Homework 5 Solutions

Homework 5 Solutions 4.2: 2: a. 321 = 256 + 64 + 1 = (01000001) 2 b. 1023 = 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = (1111111111) 2. Note that this is 1 less than the next power of 2, 1024, which

MATH 110 Spring 2015 Homework 6 Solutions

MATH 110 Spring 2015 Homework 6 Solutions Section 2.6 2.6.4 Let α denote the standard basis for V = R 3. Let α = {e 1, e 2, e 3 } denote the dual basis of α for V. We would first like to show that β =

The last three chapters introduced three major proof techniques: direct,

CHAPTER 7 Proving Non-Conditional Statements The last three chapters introduced three major proof techniques: direct, contrapositive and contradiction. These three techniques are used to prove statements

Pythagorean Quadruples 1. Pythagorean and Primitive Pythagorean Quadruples A few years ago, I became interested with those 3 vectors with integer coordinates and integer length or norm. The simplest example

ALGORITHMS FOR ALGEBRAIC CURVES

ALGORITHMS FOR ALGEBRAIC CURVES SUMMARY OF LECTURE 4 1. SCHOOF S ALGORITHM Let K be a finite field with q elements. Let p be its characteristic. Let X be an elliptic curve over K. To simplify we assume

On the generation of elliptic curves with 16 rational torsion points by Pythagorean triples

On the generation of elliptic curves with 16 rational torsion points by Pythagorean triples Brian Hilley Boston College MT695 Honors Seminar March 3, 2006 1 Introduction 1.1 Mazur s Theorem Let C be a

Lecture 3: Finding integer solutions to systems of linear equations

Lecture 3: Finding integer solutions to systems of linear equations Algorithmic Number Theory (Fall 2014) Rutgers University Swastik Kopparty Scribe: Abhishek Bhrushundi 1 Overview The goal of this lecture

The Locker Problem. Bruce Torrence and Stan Wagon

The Locker Problem Bruce Torrence and Stan Wagon The locker problem appears frequently in both the secondary and university curriculum [2, 3]. In November 2005, it appeared on National Public Radio s Car

(x + a) n = x n + a Z n [x]. Proof. If n is prime then the map

22. A quick primality test Prime numbers are one of the most basic objects in mathematics and one of the most basic questions is to decide which numbers are prime (a clearly related problem is to find

DIFFERENTIABILITY OF COMPLEX FUNCTIONS. Contents

DIFFERENTIABILITY OF COMPLEX FUNCTIONS Contents 1. Limit definition of a derivative 1 2. Holomorphic functions, the Cauchy-Riemann equations 3 3. Differentiability of real functions 5 4. A sufficient condition

CONTINUED FRACTIONS AND FACTORING. Niels Lauritzen

CONTINUED FRACTIONS AND FACTORING Niels Lauritzen ii NIELS LAURITZEN DEPARTMENT OF MATHEMATICAL SCIENCES UNIVERSITY OF AARHUS, DENMARK EMAIL: niels@imf.au.dk URL: http://home.imf.au.dk/niels/ Contents

Mathematical Induction

Mathematical Induction (Handout March 8, 01) The Principle of Mathematical Induction provides a means to prove infinitely many statements all at once The principle is logical rather than strictly mathematical,

Notes on Factoring. MA 206 Kurt Bryan

The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor

Applications of Linear Algebra to Coding Theory. Presented by:- Prof.Surjeet kaur Dept of Mathematics SIES College.Sion(W)

Applications of Linear Algebra to Coding Theory Presented by:- Prof.Surjeet kaur Dept of Mathematics SIES College.Sion(W) Outline Introduction AIM Coding Theory Vs Cryptography Coding Theory Binary Symmetric

k, then n = p2α 1 1 pα k

Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square

TAKE-AWAY GAMES. ALLEN J. SCHWENK California Institute of Technology, Pasadena, California INTRODUCTION

TAKE-AWAY GAMES ALLEN J. SCHWENK California Institute of Technology, Pasadena, California L INTRODUCTION Several games of Tf take-away?f have become popular. The purpose of this paper is to determine the

Collinear Points in Permutations

Collinear Points in Permutations Joshua N. Cooper Courant Institute of Mathematics New York University, New York, NY József Solymosi Department of Mathematics University of British Columbia, Vancouver,

Au = = = 3u. Aw = = = 2w. so the action of A on u and w is very easy to picture: it simply amounts to a stretching by 3 and 2, respectively.

Chapter 7 Eigenvalues and Eigenvectors In this last chapter of our exploration of Linear Algebra we will revisit eigenvalues and eigenvectors of matrices, concepts that were already introduced in Geometry

Systems of Linear Equations

Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and

Diagonalisation. Chapter 3. Introduction. Eigenvalues and eigenvectors. Reading. Definitions

Chapter 3 Diagonalisation Eigenvalues and eigenvectors, diagonalisation of a matrix, orthogonal diagonalisation fo symmetric matrices Reading As in the previous chapter, there is no specific essential

Integer Factorization using the Quadratic Sieve

Integer Factorization using the Quadratic Sieve Chad Seibert* Division of Science and Mathematics University of Minnesota, Morris Morris, MN 56567 seib0060@morris.umn.edu March 16, 2011 Abstract We give

Lecture 3: Linear Programming Relaxations and Rounding

Lecture 3: Linear Programming Relaxations and Rounding 1 Approximation Algorithms and Linear Relaxations For the time being, suppose we have a minimization problem. Many times, the problem at hand can

SECRET sharing schemes were introduced by Blakley [5]

206 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 52, NO. 1, JANUARY 2006 Secret Sharing Schemes From Three Classes of Linear Codes Jin Yuan Cunsheng Ding, Senior Member, IEEE Abstract Secret sharing has

MATH 4330/5330, Fourier Analysis Section 11, The Discrete Fourier Transform

MATH 433/533, Fourier Analysis Section 11, The Discrete Fourier Transform Now, instead of considering functions defined on a continuous domain, like the interval [, 1) or the whole real line R, we wish

CHAPTER 3 Numbers and Numeral Systems

CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,

Introduction to Algebraic Coding Theory

Introduction to Algebraic Coding Theory Supplementary material for Math 336 Cornell University Sarah A. Spence Contents 1 Introduction 1 2 Basics 2 2.1 Important code parameters..................... 4

minimal polyonomial Example

Minimal Polynomials Definition Let α be an element in GF(p e ). We call the monic polynomial of smallest degree which has coefficients in GF(p) and α as a root, the minimal polyonomial of α. Example: We

ON THE COMPLEXITY OF THE GAME OF SET. {kamalika,pbg,dratajcz,hoeteck}@cs.berkeley.edu

ON THE COMPLEXITY OF THE GAME OF SET KAMALIKA CHAUDHURI, BRIGHTEN GODFREY, DAVID RATAJCZAK, AND HOETECK WEE {kamalika,pbg,dratajcz,hoeteck}@cs.berkeley.edu ABSTRACT. Set R is a card game played with a

1 Eigenvalues and Eigenvectors

Math 20 Chapter 5 Eigenvalues and Eigenvectors Eigenvalues and Eigenvectors. Definition: A scalar λ is called an eigenvalue of the n n matrix A is there is a nontrivial solution x of Ax = λx. Such an x

Math 55: Discrete Mathematics

Math 55: Discrete Mathematics UC Berkeley, Spring 2012 Homework # 9, due Wednesday, April 11 8.1.5 How many ways are there to pay a bill of 17 pesos using a currency with coins of values of 1 peso, 2 pesos,

MODULAR ARITHMETIC. a smallest member. It is equivalent to the Principle of Mathematical Induction.

MODULAR ARITHMETIC 1 Working With Integers The usual arithmetic operations of addition, subtraction and multiplication can be performed on integers, and the result is always another integer Division, on

IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction

IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible

Quotient Rings and Field Extensions

Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.

Geometric models of the card game SET

Geometric models of the card game SET Cherith Tucker April 17, 2007 Abstract The card game SET can be modeled by four-dimensional vectors over Z 3. These vectors correspond to points in the affine four-space

PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient