# Cosets and Cayley-Sudoku Tables

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Cosets and Cayley-Sudoku Tables Jennifer Carmichael Chemeketa Community College Salem, OR Keith Schloeman Oregon State University Corvallis, OR Michael B Ward Western Oregon University Monmouth, OR The wildly popular Sudoku puzzles [2] are 9 9 arrays divided into nine 3 3 sub-arrays or blocks Digits 1 through 9 appear in some of the entries Other entries are blank The goal is to fill the blank entries with the digits 1 through 9 in such a way that each digit appears exactly once in each row and in each column, and in each block Table 1 gives an example of a completed Sudoku puzzle One proves in introductory group theory that every element of any group appears exactly once in each row and once in each column of the group s operation or Cayley table (In other words, any Cayley table is a Latin square) Thus, every Cayley table has two-thirds of the properties of a Sudoku table; only the subdivision of the table into blocks that contain each element exactly once is in doubt A question naturally leaps to mind: When and how can a Cayley table be arranged in such a way as to satisfy the additional requirements of being a Sudoku table? To be more specific, group elements labeling the rows and the columns of a Cayley table may be arranged in any order Moreover, in defiance of convention, row labels and column labels need not be in the same order Again we ask, when and how can the row and 1

2 Table 1: A Completed Sudoku Puzzle column labels be arranged so that the Cayley table has blocks containing each group element exactly once? For example, Table 2 shows that the completed Sudoku puzzle in Table 1 is actually a Cayley table of Z 9 := {1, 2, 3, 4, 5, 6, 7, 8, 9} under addition modulo 9 (We use 9 instead of the usual 0 in order to maintain the Sudokulike appearance) As a second example, consider A 4, the alternating group on 4 symbols We seek to arrange its elements as row and column labels so that the resulting Cayley table forms a Sudoku-like table, one in which the table is subdivided into blocks such that each group element appears exactly once in each block (as well as exactly once in each column and in each row, which, as noted, is always so in a Cayley table) Table 3 shows such an arrangement with 6 2 blocks (In constructing the table, we operate with the row label on the left and column label on the right Permutations are composed right to left For example, the entry in row (14)(23), column (134) is (14)(23)(134) = (123)) We say Tables 2 and 3 are Cayley-Sudoku tables of Z 9 and A 4, respectively In general, a Cayley-Sudoku table of a finite group G is a Cayley table for G subdivided into uniformly sized rectangular blocks in such a way that each group element appears exactly once in each block 2

3 Table 2: A Cayley Table of Z 9 with Sudoku Properties (1) (12)(34) (13)(24) (14)(23) (123) (243) (142) (134) (132) (143) (234) (124) (1) (1) (12)(34) (13)(24) (14)(23) (123) (243) (142) (134) (132) (143) (234) (124) (13)(24) (13)(24) (14)(23) (1) (12)(34) (142) (134) (123) (243) (234) (124) (132) (143) (123) (123) (134) (243) (142) (132) (124) (143) (234) (1) (14)(23) (12)(34) (13)(2 (243) (243) (142) (123) (134) (143) (234) (132) (124) (12)(34) (13)(24) (1) (14)(2 (132) (132) (234) (124) (143) (1) (13)(24) (14)(23) (12)(34) (123) (142) (134) (243) (143) (143) (124) (234) (132) (12)(34) (14)(23) (13)(24) (1) (243) (134) (142) (123) (12)(34) (12)(34) (1) (14)(23) (13)(24) (243) (123) (134) (142) (143) (132) (124) (234) (14)(23) (14)(23) (13)(24) (12)(34) (1) (134) (142) (243) (123) (124) (234) (143) (132) (134) (134) (123) (142) (243) (124) (132) (234) (143) (14)(23) (1) (13)(24) (12)(3 (142) (142) (243) (134) (123) (234) (143) (124) (132) (13)(24) (12)(34) (14)(23) (1) (234) (234) (132) (143) (124) (13)(24) (1) (12)(34) (14)(23) (142) (123) (243) (134) (124) (124) (143) (132) (234) (14)(23) (12)(34) (1) (13)(24) (134) (243) (123) (142) Table 3: A Cayley Table of A 4 with Sudoku Properties Uninteresting Cayley-Sudoku tables can be made from any Cayley table of any group by simply defining the blocks to be the individual rows (or columns) of the table Our goal in this note is to give three methods for producing interesting tables using cosets, thereby uncovering new applications of cosets (See any introductory group theory text for a review of cosets, for example, [1, Chapter 7]) Cosets Revisit Table 2 The cyclic subgroup generated by 3 in Z 9 is 3 = {9, 3, 6} The right and left cosets of 3 in Z 9 are 3 +9 = {9, 3, 6} = 9+ 3, 3

4 3 + 1 = {1, 4, 7} = 1 + 3, and = {2, 5, 8} = With only a little prompting, we quickly see that the columns in each block are labeled by elements of right cosets of 3 in Z 9 Each set of elements labeling the rows of a block contains exactly one element from each left coset Equivalently, the row labels partition Z 9 into complete sets of left coset representatives of 3 in Z 9 (Momentarily we shall see why we bothered to distinguish between right and left) Reexamining Table 3 reveals a similar structure Consider the subgroup H := (12)(34) = {(1), (12)(34)} We brush-up on composing permutations (right to left) by calculating the right coset H(123) = {(1)(123), (12)(34)(123)} = {(123), (243)} and the corresponding left coset {(123)(1), (123)(12)(34)} = {(123), (134)} In that fashion, we find the right cosets to be H(1) = {(1), (12)(34)}, H(13)(24) = {(13)(24), (14)(23)}, H(123) = {(123), (243)}, H(142) = {(142), (134)}, H(132) = {(132), (143)}, and H(234) = {(234), (124)} while the left cosets are (1)H = {(1), (12)(34)}, (13)(24)H = {(13)(24), (14)(23)}, (123)H = {(123), (134)}, (243)H = {(243), (142)}, (132)H = {(132), (234)}, (143)H = {(143), (124)} This time we know what to expect Sure enough, the columns in Table 3 are labeled by the elements of the distinct right cosets of H in A 4 while the row labels partition A 4 into complete sets of left coset representatives of H in A 4 Those examples illustrate our first general construction Before proceeding, let us agree upon a convention for labeling a Cayley table When a set is listed in a row or column of the table, it is to be interpreted as the individual elements of that set being listed in separate rows or columns, respectively For example, under that convention, the rows and columns of Table 2 could be labeled {9, 3, 6} {1, 4, 7} {9, 1, 2} {3, 4, 5} where the label {9, 1, 2} is interpreted as the elements 9, 1, and 2 listed 4

5 vertically, one per row, and {9, 3, 6} is interpreted as the elements 9, 3, and 6 listed horizontally, one per column Cayley-Sudoku Construction 1 Let G be a finite group Assume H is a subgroup of G having order k and index n (so that G = nk) If Hg 1, Hg 2,, Hg n are the n distinct right cosets of H in G, then arranging the Cayley table of G with columns labeled by the cosets Hg 1, Hg 2,, Hg n and the rows labeled by sets T 1, T 2,, T k (as in Table 4) yields a Cayley-Sudoku table of G with blocks of dimension n k if and only if T 1, T 2,, T k partition G into complete sets of left coset representatives of H in G T 1 T 2 T k Hg 1 Hg 2 Hg n Table 4: Construction 1 Using Right Cosets and Left Coset Representatives Furthermore, if y 1 H, y 2 H,, y n H are the n distinct left cosets of H in G, then arranging the Cayley table of G with rows labeled by the cosets y 1 H, y 2 H,, y n H and the columns labeled by sets R 1, R 2,, R k yields a Cayley-Sudoku table of G with blocks of dimension k n if and only if R 1, R 2,, R k partition G into complete sets of right coset representatives of H in G Note that the second version of the Construction 1 is dual to the first, obtained by reversing the left with right and rows with columns Now let us prove the correctness of the construction for the case of right cosets An arbitrary block of the table, indexed by T h = {t 1, t 2,, t n } and Hg i, is the given the following table 5

6 Hg i t 1 t 2 t n t 1 Hg i t 2 Hg i t n Hg i The elements in the block are the elements of the set B := t 1 Hg i t 2 Hg i t n Hg i = (t 1 H t 2 H t n H)g i, the equality being a routine exercise We want to show that the elements of G appear exactly once in that block if and only if T h is a complete set of left coset representatives of H in G If T h is a complete set of left coset representatives, then t 1 H t 2 H t n H = G So we have B = Gg i = G Thus every element of G appears in every block But the number of entries in the block is nk and the order of G is nk, so every element of G appears exactly once in each block On the other hand, if every element appears exactly once in each block, then B = G and that gives us G = Gg 1 i = Bg 1 i = t 1 H t 2 H t n H There are n cosets in this union, each having order k Therefore, since the union of those cosets is the entire group G and G = nk, we must have n distinct cosets in the union Thus, {t 1, t 2,, t n } is a complete set of left coset representatives of H in G, as claimed The first punch line of this section is that any proper non-trivial subgroup of a finite group gives rise to interesting Cayley-Sudoku tables by using Construction 1 Therefore, any finite group having a proper non-trivial subgroup (which is to say, any finite group whose order is not a prime) admits an interesting Cayley-Sudoku table We now introduce another construction utilizing cosets This construction is like-handed, that is, it uses left cosets and left coset representatives (or right and right) Construction 1 is cross-handed It uses right cosets and left coset representatives or left cosets and right coset representatives For this construction, we recall a standard group theoretic definition For any subgroup H of a group G and for any g G, g 1 Hg := {g 1 hg : h H} is called a conjugate of H and is denoted H g It is routine to show H g is a 6

7 subgroup of G Cayley-Sudoku Construction 2 Assume H is a subgroup of G having order k and index n Also suppose t 1 H, t 2 H,, t n H are the distinct left cosets of H in G Arranging the Cayley table of G with columns labeled by the cosets t 1 H, t 2 H,, t n H and the rows labeled by sets L 1, L 2,, L k yields a Cayley-Sudoku table of G with blocks of dimension n k if and only if L 1, L 2,, L k are complete sets of left coset representatives of H g for all g G L 1 L 2 L k t 1 H t 2 H t n H Table 5: Construction 2 Using Left Cosets and Left Coset Representatives Since one or two interesting subtleties arise, we will verify the correctness of Construction 2 Consider an arbitrary block in the table, indexed by L i := {g i1, g i2, g in } and t j H Note, however, that t j H = t j Ht 1 j t j = H t 1 j t j Thus, our block is the following H t 1 j t j g i1 g i2 g in g i1 H t 1 j t j g i2 H t 1 j t j g in H t 1 j t j The set of elements in the block is g i1 H t 1 j (g i1 H t 1 j g in H t 1 j )t j 7 t j g i2 H t 1 j t j g 1n H t 1 j t j =

8 Suppose L 1, L 2,, L k are complete sets of left coset representatives of H g for all g G, then L i = {g i1, g i2,, g in } is a complete set of left coset representatives of H t 1 j Therefore, just as in the verification of Construction 1, we can show every element of G appears exactly once in the block Conversely, suppose every element of G appears exactly once in each block Once again arguing as in Construction 1, we conclude each L i is a complete set of left coset representatives of H t j for every j In order finish, we need a (known) result of independent group theoretic interest Namely, with notation as in the Construction, for every g G, there exists t j such that H g = H t 1 j To see that, let g G, then g 1 is in some left coset of H, say t j H Thus, g 1 = t j h for some h H Armed with the observation hhh 1 = H (easily shown since H is a subgroup), we hit our target: H g = g 1 Hg = (t j h)h(h 1 t 1 j ) = t j (hhh 1 )t 1 = t j Ht 1 j = H t 1 j Combining this with the preceding paragraph, we can conclude L 1, L 2,, L k are complete sets of left coset representatives of H g for all g G, as claimed We invite the reader to formulate and verify a right-handed version of Construction 2 We also raise an interesting and, evidently, non-trivial question for further investigation Under what circumstances can one decompose a finite group G in the way required by Construction 2? There is one easy circumstance If H is a normal subgroup of a G, then it is not difficult to show H g = H for every g G [1, Chapter 9] Thus, decomposing G into complete sets of left coset representatives of H will do the trick Sadly, in that case, Construction 2 gives the same Cayley-Sudoku table as Construction 1 because the left cosets indexing the columns equal the corresponding right cosets by normality Happily, we know of one general circumstance in which we can decompose G in the desired way to obtain new Cayley-Sudoku tables It is contained in the following proposition, stated without proof Proposition Suppose the finite group G contains subgroups T := {t 1, t 2,, t n } and H := {h 1, h 2,, h k } such that G = {th : t T, h H} := T H and T H = {e}, then the elements of T form a complete set of left coset repre- 8

9 sentatives of H and the cosets T h 1, T h 2,, T h k decompose G into complete sets of left coset representatives of H g for every g G In other words, from the Proposition, Construction 2 applies when we set L i := T h i and use the left cosets t 1 H, t 2 H,, t n H Let us try it out on the group S 4 Let H = (123) = {(1), (123), (132)} and T = {(1), (12)(34), (13)(24), (14)(23), (24), (1234), One can check (by brute force, if necessary) that H and T are subgroups of S 4 satisfying the hypotheses of the Proposition Therefore, according to Construction 2, the following table yields a Cayley-Sudoku table of S 4 H (12)(34)H (13)(24)H (14)(23)H (24)H (1234)H (1432)H (13)H T T (123) T (132) Seeking to be convinced that is a new Cayley-Sudoku table, not of the kind produced by Construction 1, we examine the sets indexing the columns and rows In Construction 1, the sets indexing columns are right cosets of some subgroup or else the sets indexing the rows are left cosets of some subgroup In our table, the only subgroup indexing the columns is H and most of the remaining index sets are not right cosets of H For example, (12)(34)H H(12)(34) and so it is not a right coset of H Similar consideration of the sets indexing the rows shows Construction 1 was not on the job here Extending Cayley-Sudoku tables Our final construction shows a way to extend a Cayley-Sudoku table of a subgroup to a Cayley-Sudoku table of the full group Cayley-Sudoku Construction 3 Let G be a finite group with a subgroup A Let C 1, C 2,, C k partition A and R 1, R 2, R n partition A such that the following table is a Cayley-Sudoku table of A 9

10 C 1 C 2 C k R 1 R 2 R n If {l 1, l 2,, l t } and {r 1, r 2, r t } are complete sets of left and right coset representatives, respectively, of A in G, then arranging the Cayley table of G with columns labeled with the sets C i r j, i = 1,, k, j = 1,, t and the b th block of rows labeled with l j R b, j = 1,, t, for b = 1,, n (as in Table 6) yields a Cayley-Sudoku table of G with blocks of dimension tk n C 1 r 1 C 2 r 1 C k r 1 C 1 r 2 C k r 2 C 1 r t C k r t l 1 R 1 l 2 R 1 l t R 1 l 1 R 2 l t R 2 l 1 R n l t R n Table 6: Construction 3 Proving the correctness of Construction 3 is quite like the proof for Construction 1 We leave it as an exercise for the reader and proceed to an example Working in the group Z 8 := {0, 1, 2, 3, 4, 5, 6, 7} under addition modulo 8, let us apply Construction 1 to form a Cayley-Sudoku table for the 10

11 subgroup 2 and then extend that table to a Cayley-Sudoku table of Z 8 via Construction 3 Observe that 4 = {0, 4} is a subgroup of 2 = {0, 2, 4, 6} The left and right cosets of 4 in 2 are = {0, 4} = and = {2, 6} = Thus, {0, 2} and {4, 6} partition Z 8 into complete sets of right coset representatives Applying Construction 1, wherein elements of left cosets label the rows and right coset representatives label the columns, yields Table Table 7: Construction 1 Applied Now the left and right cosets of 2 in Z 8 are 0+ 2 = {0, 2, 4, 6} = 2 +0 and = {1, 3, 5, 7} = Accordingly, {0, 1} is a complete set of left and right coset representatives of 2 in Z 8 According to Construction 3, Table 8 should be (and is, much to our relief) a Cayley-Sudoku table of Z 8 For easy comparison with Table 6, rows and columns are labeled both with sets and with individual elements We chose Z 8 for our example because it has enough subgroups to make Construction 3 interesting, yet the calculations are easy to do and the resulting table fits easily on a page In one sense, however, the calculations are too easy Since Z 8 is abelian, all the corresponding right and left cosets of any subgroup are equal (In other words, all the subgroups are normal) Thus, the role of right versus left in Construction 3 is obscured The interested reader may wish to work out an example where right and left cosets are different For instance, in S 4, one could consider the subgroup A := {(1), (12)(34), (13)(24), (14)(23), (24), (1234), (1432), (13)} Use Con- 11

12 0 + {0, 4} 1 + {0, 4} 0 + {2, 6} 1 + {2, 6} {0, 2} + 0 {0, 2} + 1 {4, 6} + 0 {4, 6} Table 8: A Cayley-Sudoku Table of Z 8 from Construction 3 struction 1 with the subgroup (24) of A to obtain a Cayley-Sudoku table of A, then apply Construction 3 to extend that table to a Cayley-Sudoku table of S 4 The associated computations are manageable (barely, one might think by the end!) and the roles of right and left are more readily apparent Table 8 is a new sort of Cayley-Sudoku table, one not produced by either of Constructions 1 or 2 To see why, recall that in Construction 1 and 2 (including the right-handed cousin of 2), either the columns or the rows in the blocks are labeled by cosets of a subgroup One of those cosets is, of course, the subgroup itself However, we easily check that none of the sets labeling columns or rows of the blocks in Table 8 are subgroups of Z 8 A puzzle The ubiquitous Sudoku leads many students to treat the familiar exercise of filling in the missing entries of a partial Cayley table as a special sort of Sudoku puzzle In a recent group theory course taught by the third author, several students explained how they deduced missing entries in such an exercise [1, exercise 25 p 55] by writing I Sudokued them Meaning they applied Sudoku-type logic based on the fact that rows and columns of a Cayley table contain no repeated entries 12

13 We extend that notion by including a Cayley-Sudoku puzzle for the reader It requires both group theoretic and Sudoku reasoning The group theory required is very elementary (In particular, one need not use the classification of groups of order 8) The puzzle has three parts, one for entertainment and two to show this is truly a new sort of puzzle First, complete Table 9 with 2 4 blocks as indicated so that it becomes a Cayley-Sudoku table Do not assume a priori that Table 9 was produced by any of Constructions 1-3 Second, show group theoretic reasoning is actually needed in the puzzle by completing Table 9 so that it satisfies the three Sudoku properties for the indicated 2 4 blocks but is not the Cayley table of any group Third, show Sudoku reasoning is required by finding another way to complete Table 9 so that it is a Cayley table of some group, but not a Cayley-Sudoku table Table 9: A Cayley-Sudoku Puzzle 13

14 Ideas for further study By exhaustive (in more ways than one) analysis of cases, the authors can show that the only 9 9 Cayley-Sudoku tables are those resulting from Construction 1 Is the same true for p 2 p 2 Cayley- Sudoku tables where p is a prime? All the constructions of Cayley-Sudoku tables known to the authors, including some not presented in this paper, ultimately rely on cosets and coset representatives Are there Cayley-Sudoku constructions that do not use cosets and coset representatives? Related to the previous question, how does one create a single block of a Cayley-Sudoku table? That is, if G is a group with subsets (not necessarily subgroups) K and H such that G = K H, what are nice conditions under which we will have KH = G? Can a Cayley-Sudoku table of a group be used to construct a Cayley- Sudoku table of a subgroup or a factor group? Can a Cayley-Sudoku table of a factor group be used to construct a Cayley-Sudoku table of the original group? Are there efficient algorithms for generating interesting Cayley-Sudoku puzzles? Making the definition of Cayley-Sudoku tables less restrictive can lead to some interesting examples For instance if the definition of Cayley-Sudoku tables is altered so that the individual blocks of the table do not have to be of fixed dimension we obtain in Table 10 an example of a generalized Cayley-Sudoku table of the group Z 8 What is a construction method for such generalized Cayley-Sudoku tables? How about for jigsaw Cayley-Sudoku tables wherein the blocks are not rectangles? Perhaps most interesting of all, find other circumstances under which Construction 2 applies Acknowledgement This note is an outgrowth of the senior theses of the first two authors written under the supervision of the third author We 14

15 Table 10: A Generalized Cayley-Sudoku Table thank one another for many hours of satisfying and somewhat whimsical mathematics We also thank the students at DigiPen Institute of Technology, where the first author was a guest speaker, for suggesting that we make a Cayley-Sudoku puzzle References 1 JA Gallian, Contemporary Abstract Algebra, 6th ed, Houghton Mifflin Co, New York, NY, R Wilson, The Sudoku Epidemic, FOCUS 26 (2006), 5-7 Puzzle Solutions face for easy identification In each solution, the original puzzle entries are in bold Part 1 Table 11 shows the solution Clearly, it satisfies the three Sudoku properties We will show it is the Cayley table of D 4, the dihedral group of order 8 We will regard D 4 as the group of symmetries of a square Let R 90 be a counterclockwise rotation about the center of the square and let H be a reflection across a line through the center of the square that is parallel to a side of the square The eight elements of D 4 are R 0 90, R 1 90, R 2 90, R 3 90, HR 0 90, 15

16 HR90, 1 HR90, 2 and HR90 3 Numbering those elements 1 through 8 in the order given and then calculating the Cayley table gives Table 11 Thus, we have a Cayley-Sudoku table as claimed By the way, it was obtained by applying Construction 1 to the subgroup R Table 11: Cayley-Sudoku Puzzle Solution Part 2 Table 12 visibly satisfies the Sudoku conditions It is not a Cayley table For otherwise, 1 7 = 7 implies 1 is the identity However, Table 12: Sudoku-not-Cayley Puzzle Solution 16

17 Part 3 Table 13 does not satisfy the Sudoku conditions Blocks contain repeated entries It is, however, a Cayley table One can check that it is again the Cayley table of D 4 Just change the labeling of R90 2 from 3 to 5 and of H from 5 to 3 Table 13 is the recalculated Cayley table Table 13: Cayley-not-Sudoku Puzzle Solution 17

### Cayley-Sudoku Tables, Quasigroups, and More Questions. Undergraduate Research

Cayley-Sudoku Tables, Loops, Quasigroups, and More Questions from Undergraduate Research XXXI Ohio State-Denison Mathematics Conference May 2012 First Cayley Table (1854) & First Sudoku Puzzle (1979) &

### ORDERS OF ELEMENTS IN A GROUP

ORDERS OF ELEMENTS IN A GROUP KEITH CONRAD 1. Introduction Let G be a group and g G. We say g has finite order if g n = e for some positive integer n. For example, 1 and i have finite order in C, since

### Counting Magic Cayley-Sudoku Tables

Counting Magic Cayley-Sudoku Tables Michael Ward Western Oregon University Zassenhaus Group Theory Conference May 2015 In memory of Wolfgang Kappe 1 / 48 Outline Counting Magic Sudoku Tables for Z 9 [Lorch

### 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

### To introduce the concept of a mathematical structure called an algebraic group. To illustrate group concepts, we introduce cyclic and dihedral groups.

1 Section 6. Introduction to the Algebraic Group Purpose of Section: To introduce the concept of a mathematical structure called an algebraic group. To illustrate group concepts, we introduce cyclic and

### An Investigation of Erd s Method: A Scheme for Generating Carmichael Numbers

An Investigation of Erd s Method: A Scheme for Generating Carmichael Numbers By: Trevor Brennan Advisor: John Greene 1 Contents Chapter 1: Introduction. 1 Chapter 2: Number and Group Theory Principles.

### Elements of Abstract Group Theory

Chapter 2 Elements of Abstract Group Theory Mathematics is a game played according to certain simple rules with meaningless marks on paper. David Hilbert The importance of symmetry in physics, and for

### COLORED GRAPHS AND THEIR PROPERTIES

COLORED GRAPHS AND THEIR PROPERTIES BEN STEVENS 1. Introduction This paper is concerned with the upper bound on the chromatic number for graphs of maximum vertex degree under three different sets of coloring

### 2. Methods of Proof Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try.

2. METHODS OF PROOF 69 2. Methods of Proof 2.1. Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try. Trivial Proof: If we know q is true then

### Sets and functions. {x R : x > 0}.

Sets and functions 1 Sets The language of sets and functions pervades mathematics, and most of the important operations in mathematics turn out to be functions or to be expressible in terms of functions.

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

### 13 Solutions for Section 6

13 Solutions for Section 6 Exercise 6.2 Draw up the group table for S 3. List, giving each as a product of disjoint cycles, all the permutations in S 4. Determine the order of each element of S 4. Solution

### Combinatorial Concepts With Sudoku I: Symmetry

ombinatorial oncepts With Sudoku I: Symmetry Gordon Royle March 29, 2006 (Version 0.2) bstract The immensely popular logic game Sudoku incorporates a significant number of combinatorial concepts. In this

### S on n elements. A good way to think about permutations is the following. Consider the A = 1,2,3, 4 whose elements we permute with the P =

Section 6. 1 Section 6. Groups of Permutations: : The Symmetric Group Purpose of Section: To introduce the idea of a permutation and show how the set of all permutations of a set of n elements, equipped

### GROUP ACTIONS ON SETS WITH APPLICATIONS TO FINITE GROUPS

GROUP ACTIONS ON SETS WITH APPLICATIONS TO FINITE GROUPS NOTES OF LECTURES GIVEN AT THE UNIVERSITY OF MYSORE ON 29 JULY, 01 AUG, 02 AUG, 2012 K. N. RAGHAVAN Abstract. The notion of the action of a group

### Sudoku puzzles and how to solve them

Sudoku puzzles and how to solve them Andries E. Brouwer 2006-05-31 1 Sudoku Figure 1: Two puzzles the second one is difficult A Sudoku puzzle (of classical type ) consists of a 9-by-9 matrix partitioned

### Mathematics of Sudoku I

Mathematics of Sudoku I Bertram Felgenhauer Frazer Jarvis January, 00 Introduction Sudoku puzzles became extremely popular in Britain from late 00. Sudoku, or Su Doku, is a Japanese word (or phrase) meaning

### Permutation Groups. Rubik s Cube

Permutation Groups and Rubik s Cube Tom Davis tomrdavis@earthlink.net May 6, 2000 Abstract In this paper we ll discuss permutations (rearrangements of objects), how to combine them, and how to construct

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +

### (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

### Cracking the Sudoku: A Deterministic Approach

Cracking the Sudoku: A Deterministic Approach David Martin Erica Cross Matt Alexander Youngstown State University Center for Undergraduate Research in Mathematics Abstract The model begins with the formulation

### You ve seen them played in coffee shops, on planes, and

Every Sudoku variation you can think of comes with its own set of interesting open questions There is math to be had here. So get working! Taking Sudoku Seriously Laura Taalman James Madison University

### = 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

### Congruences. Robert Friedman

Congruences Robert Friedman Definition of congruence mod n Congruences are a very handy way to work with the information of divisibility and remainders, and their use permeates number theory. Definition

### ALGEBRA HANDOUT 2: IDEALS AND QUOTIENTS. 1. Ideals in Commutative Rings In this section all groups and rings will be commutative.

ALGEBRA HANDOUT 2: IDEALS AND QUOTIENTS PETE L. CLARK 1. Ideals in Commutative Rings In this section all groups and rings will be commutative. 1.1. Basic definitions and examples. Let R be a (commutative!)

### Investigations into the Card Game SET THE MAXIMUM NUMBER OF SETS FOR N CARDS AND THE TOTAL NUMBER OF INTERNAL SETS FOR ALL PARTITIONS OF THE DECK

Investigations into the Card Game SET THE MAXIMUM NUMBER OF SETS FOR N CARDS AND THE TOTAL NUMBER OF INTERNAL SETS FOR ALL PARTITIONS OF THE DECK By Jim Vinci June 30, 2009 E-mail address:mathisfun@wowway.com

### SOLUTIONS FOR PROBLEM SET 6

SOLUTIONS FOR PROBLEM SET 6 A. Suppose that G is a group and that H is a subgroup of G such that [G : H] = 2. Suppose that a,b G, but a H and b H. Prove that ab H. SOLUTION: Since [G : H] = 2, it follows

### Chapter 7. Permutation Groups

Chapter 7 Permutation Groups () We started the study of groups by considering planar isometries In the previous chapter, we learnt that finite groups of planar isometries can only be cyclic or dihedral

### Sudoku, logic and proof Catherine Greenhill

Sudoku, logic and proof Catherine Greenhill As you probably know, a Sudoku puzzle is a 9 9 grid divided into nine 3 3 subgrids. Some of the cells in the grid contain a symbol: usually the symbols are the

### Geometric Group Theory Homework 1 Solutions. e a b e e a b a a b e b b e a

Geometric Group Theory Homework 1 Solutions Find all groups G with G = 3 using multiplications tables and the group axioms. A multiplication table for a group must be a Latin square: each element appears

### Chapter 3. Cartesian Products and Relations. 3.1 Cartesian Products

Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing

### Enumerating possible Sudoku grids

Enumerating possible Sudoku grids Bertram Felgenhauer Department of Computer Science TU Dresden 00 Dresden Germany bf@mail.inf.tu-dresden.de Frazer Jarvis Department of Pure Mathematics University of Sheffield,

### Chapter Three. Functions. In this section, we study what is undoubtedly the most fundamental type of relation used in mathematics.

Chapter Three Functions 3.1 INTRODUCTION In this section, we study what is undoubtedly the most fundamental type of relation used in mathematics. Definition 3.1: Given sets X and Y, a function from X to

### 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,

### 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

### Creating Clueless Puzzles Gerard Butters Frederick Henle James Henle Colleen McGaughey 1

Creating Clueless Puzzles Gerard Butters Frederick Henle James Henle Colleen McGaughey This is an invitation and a report. The report is of our efforts to create puzzles of a special kind. The invitation

### 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

### Permutation Groups. Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles April 2, 2003

Permutation Groups Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles April 2, 2003 Abstract This paper describes permutations (rearrangements of objects): how to combine them, and how

### INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS

INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS STEVEN HEILMAN Contents 1. Homework 1 1 2. Homework 2 6 3. Homework 3 10 4. Homework 4 16 5. Homework 5 19 6. Homework 6 21 7. Homework 7 25 8. Homework 8 28

### Group Theory (MA343): Lecture Notes Semester I Dr Rachel Quinlan School of Mathematics, Statistics and Applied Mathematics, NUI Galway

Group Theory (MA343): Lecture Notes Semester I 2013-2014 Dr Rachel Quinlan School of Mathematics, Statistics and Applied Mathematics, NUI Galway November 21, 2013 Contents 1 What is a group? 2 1.1 Examples...........................................

### 9 abcd = dcba. 9 ( b + 10c) = c + 10b b + 90c = c + 10b b = 10c.

In this session, we ll learn how to solve problems related to place value. This is one of the fundamental concepts in arithmetic, something every elementary and middle school mathematics teacher should

### 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

### *.I Zolotareff s Proof of Quadratic Reciprocity

*.I. ZOLOTAREFF S PROOF OF QUADRATIC RECIPROCITY 1 *.I Zolotareff s Proof of Quadratic Reciprocity This proof requires a fair amount of preparations on permutations and their signs. Most of the material

### 3 Congruence arithmetic

3 Congruence arithmetic 3.1 Congruence mod n As we said before, one of the most basic tasks in number theory is to factor a number a. How do we do this? We start with smaller numbers and see if they divide

### 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

### 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

### MATH 2030: SYSTEMS OF LINEAR EQUATIONS. ax + by + cz = d. )z = e. while these equations are not linear: xy z = 2, x x = 0,

MATH 23: SYSTEMS OF LINEAR EQUATIONS Systems of Linear Equations In the plane R 2 the general form of the equation of a line is ax + by = c and that the general equation of a plane in R 3 will be we call

### 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

### Applications of Methods of Proof

CHAPTER 4 Applications of Methods of Proof 1. Set Operations 1.1. Set Operations. The set-theoretic operations, intersection, union, and complementation, defined in Chapter 1.1 Introduction to Sets are

### 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,

### 1 Symmetries of regular polyhedra

1230, notes 5 1 Symmetries of regular polyhedra Symmetry groups Recall: Group axioms: Suppose that (G, ) is a group and a, b, c are elements of G. Then (i) a b G (ii) (a b) c = a (b c) (iii) There is an

### Math 210A: Algebra, Homework 1

Math 210A: Algebra, Homework 1 Ian Coley October 9, 2013 Problem 1. Let a 1, a 2,..., a n be elements of a group G. Define the product of the a i s by induction: a 1 a 2 a n = (a 1 a 2 a n 1 )a n. (a)

### FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES

FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied

### An important class of codes are linear codes in the vector space Fq n, where F q is a finite field of order q.

Chapter 3 Linear Codes An important class of codes are linear codes in the vector space Fq n, where F q is a finite field of order q. Definition 3.1 (Linear code). A linear code C is a code in Fq n for

### RELATIONS AND FUNCTIONS

Chapter 1 RELATIONS AND FUNCTIONS There is no permanent place in the world for ugly mathematics.... It may be very hard to define mathematical beauty but that is just as true of beauty of any kind, we

### Sudoku Puzzles and How to Solve Them

1 258 NAW 5/7 nr. 4 december 2006 Sudoku Puzzles and How to Solve Them Andries E. Brouwer Andries E. Brouwer Technische Universiteit Eindhoven Postbus 513, 5600 MB Eindhoven aeb@win.tue.nl Recreational

### (Refer Slide Time: 05:02)

Discrete Mathematical Structures Dr. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture - 1 Propositional Logic This course is about discrete

### The Topology of Spaces of Triads

The Topology of Spaces of Triads Michael J. Catanzaro August 28, 2009 Abstract Major and minor triads in the diatonic scale can be naturally assembled into a space which turns out to be a torus. We study

3. QUADRATIC CONGRUENCES 3.1. Quadratics Over a Finite Field We re all familiar with the quadratic equation in the context of real or complex numbers. The formula for the solutions to ax + bx + c = 0 (where

### CLASSIFYING FINITE SUBGROUPS OF SO 3

CLASSIFYING FINITE SUBGROUPS OF SO 3 HANNAH MARK Abstract. The goal of this paper is to prove that all finite subgroups of SO 3 are isomorphic to either a cyclic group, a dihedral group, or the rotational

### SUBGROUPS OF CYCLIC GROUPS. 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by

SUBGROUPS OF CYCLIC GROUPS KEITH CONRAD 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by g = {g k : k Z}. If G = g, then G itself is cyclic, with g as a generator. Examples

### DISCRETE MATHEMATICS W W L CHEN

DISCRETE MATHEMATICS W W L CHEN c W W L Chen, 1982, 2008. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It is available free

### 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

### Chapter 7: Products and quotients

Chapter 7: Products and quotients Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 42, Spring 24 M. Macauley (Clemson) Chapter 7: Products

### Students in their first advanced mathematics classes are often surprised

CHAPTER 8 Proofs Involving Sets Students in their first advanced mathematics classes are often surprised by the extensive role that sets play and by the fact that most of the proofs they encounter are

### 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

### L(n) OA(n, 3): if L is a latin square with colour set [n] then the following is an O(n, 3):

9 Latin Squares Latin Square: A Latin Square of order n is an n n array where each cell is occupied by one of n colours with the property that each colour appears exactly once in each row and in each column.

### 1.3 Induction and Other Proof Techniques

4CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU 1.3 Induction and Other Proof Techniques The purpose of this section is to study the proof technique known as mathematical induction.

### The noblest pleasure is the joy of understanding. (Leonardo da Vinci)

Chapter 6 Back to Geometry The noblest pleasure is the joy of understanding. (Leonardo da Vinci) At the beginning of these lectures, we studied planar isometries, and symmetries. We then learnt the notion

### Unit 1 Number Sense. All other lessons in this unit are essential for both the Ontario and WNCP curricula.

Unit Number Sense In this unit, students will study factors and multiples, LCMs and GCFs. Students will compare, add, subtract, multiply, and divide fractions and decimals. Meeting your curriculum Many

### Basic Category Theory for Models of Syntax (Preliminary Version)

Basic Category Theory for Models of Syntax (Preliminary Version) R. L. Crole ( ) Department of Mathematics and Computer Science, University of Leicester, Leicester, LE1 7RH, U.K. Abstract. These notes

### Published in AMESA News, Nov. 2015, Vol. 57, pp

ORDER OF OPERATIONS: THE MATH AND THE MYTH Michael de Villiers Dept. of Mathematics and Computer Science Education University of KwaZulu-Natal, South Africa Dynamic Math site: http://dynamicmathematicslearning.com/javagsplinks.htm

### Department of Mathematics Exercises G.1: Solutions

Department of Mathematics MT161 Exercises G.1: Solutions 1. We show that a (b c) = (a b) c for all binary strings a, b, c B. So let a = a 1 a 2... a n, b = b 1 b 2... b n and c = c 1 c 2... c n, where

### 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

### ENUMERATION BY ALGEBRAIC COMBINATORICS. 1. Introduction

ENUMERATION BY ALGEBRAIC COMBINATORICS CAROLYN ATWOOD Abstract. Pólya s theorem can be used to enumerate objects under permutation groups. Using group theory, combinatorics, and many examples, Burnside

### Pythagorean Triples, Complex Numbers, Abelian Groups and Prime Numbers

Pythagorean Triples, Complex Numbers, Abelian Groups and Prime Numbers Amnon Yekutieli Department of Mathematics Ben Gurion University email: amyekut@math.bgu.ac.il Notes available at http://www.math.bgu.ac.il/~amyekut/lectures

### COMP 250 Fall Mathematical induction Sept. 26, (n 1) + n = n + (n 1)

COMP 50 Fall 016 9 - Mathematical induction Sept 6, 016 You will see many examples in this course and upcoming courses of algorithms for solving various problems It many cases, it will be obvious that

### Theorem Let G be a group and H a normal subgroup of G. Then the operation given by (Q) is well defined.

22. Quotient groups I 22.1. Definition of quotient groups. Let G be a group and H a subgroup of G. Denote by G/H the set of distinct (left) cosets with respect to H. In other words, we list all the cosets

### 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

### CONTENTS 1. Peter Kahn. Spring 2007

CONTENTS 1 MATH 304: CONSTRUCTING THE REAL NUMBERS Peter Kahn Spring 2007 Contents 2 The Integers 1 2.1 The basic construction.......................... 1 2.2 Adding integers..............................

### Binary Operations. Discussion. Power Set Operation. Binary Operation Properties. Whole Number Subsets. Let be a binary operation defined on the set A.

Binary Operations Let S be any given set. A binary operation on S is a correspondence that associates with each ordered pair (a, b) of elements of S a uniquely determined element a b = c where c S Discussion

### 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.

### COMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS

Bull Austral Math Soc 77 (2008), 31 36 doi: 101017/S0004972708000038 COMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS IGOR V EROVENKO and B SURY (Received 12 April 2007) Abstract We compute

### Mathematics of Cryptography

CHAPTER 2 Mathematics of Cryptography Part I: Modular Arithmetic, Congruence, and Matrices Objectives This chapter is intended to prepare the reader for the next few chapters in cryptography. The chapter

### CHAPTER 17. Linear Programming: Simplex Method

CHAPTER 17 Linear Programming: Simplex Method CONTENTS 17.1 AN ALGEBRAIC OVERVIEW OF THE SIMPLEX METHOD Algebraic Properties of the Simplex Method Determining a Basic Solution Basic Feasible Solution 17.2

### Handout #1: Mathematical Reasoning

Math 101 Rumbos Spring 2010 1 Handout #1: Mathematical Reasoning 1 Propositional Logic A proposition is a mathematical statement that it is either true or false; that is, a statement whose certainty or

### NUMB3RS Activity: Sudoku Puzzles. Episode: All s Fair

Teacher Page 1 NUMBRS Activity: Sudoku Puzzles Topic: Sudoku Puzzles Grade Level: 8-10 Objective: Explore sudoku puzzles and Latin squares Time: 60-90 minutes Introduction In All s Fair, Alan, Larry, and

### 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

### Discrete Mathematics and Probability Theory Fall 2016 Seshia and Walrand Note 3

CS 70 Discrete Mathematics and Probability Theory Fall 06 Seshia and Walrand Note 3 Mathematical Induction Introduction. In this note, we introduce the proof technique of mathematical induction. Induction

### Creating Sudoku Puzzles

Page 1 of 15 Control # 2883 Creating Sudoku Puzzles Control # 2883 February 19, 2008 Abstract We consider the problem of generating well-formed Sudoku puzzles. Weighing the benefits of constructing a problem

### Section 3.3 Equivalence Relations

1 Section 3.3 Purpose of Section To introduce the concept of an equivalence relation and show how it subdivides or partitions a set into distinct categories. Introduction Classifying objects and placing

### Sudoku as an Expression in Propositional Logic

Sudoku as an Expression in Propositional Logic Peter Cappello Department of Computer Science University of California, Santa Barbara INTRODUCTION Sudoku is a kind of puzzle that involves a 9 9 array of

### Definition. Proof. Notation and Terminology

ÌÖ Ò ÓÖÑ Ø ÓÒ Ó Ø ÈÐ Ò A transformation of the Euclidean plane is a rule that assigns to each point in the plane another point in the plane. Transformations are also called mappings, or just maps. Examples

### the lemma. Keep in mind the following facts about regular languages:

CPS 2: Discrete Mathematics Instructor: Bruce Maggs Assignment Due: Wednesday September 2, 27 A Tool for Proving Irregularity (25 points) When proving that a language isn t regular, a tool that is often

### A set is a Many that allows itself to be thought of as a One. (Georg Cantor)

Chapter 4 Set Theory A set is a Many that allows itself to be thought of as a One. (Georg Cantor) In the previous chapters, we have often encountered sets, for example, prime numbers form a set, domains

### 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,

### 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

### Mathematical Induction

Chapter 2 Mathematical Induction 2.1 First Examples Suppose we want to find a simple formula for the sum of the first n odd numbers: 1 + 3 + 5 +... + (2n 1) = n (2k 1). How might we proceed? The most natural