The Applied Algebra Workbook David Nacin William Paterson University March 22, 2018 2 Contents 1 Some Basic Results in Group Theory5 1.1 The Dihedral Groups.........................................5 1.1.1 Arbitrary Dihedral Group Questions............................6 1.1.2 The Group D3 ........................................8 1.1.3 The Group D4 ........................................ 11 1.1.4 The Group D5 ........................................ 14 1.1.5 The Group D6 ........................................ 17 1.2 The Symmetric Groups....................................... 21 1.2.1 The Group S3 ........................................ 23 1.2.2 The Group S4 ........................................ 26 1.2.3 The Group S5 ........................................ 30 1.2.4 Dihedral Groups as Subgroups of Symmetric Groups................... 34 1.3 Groups of Rotational Symmetries.................................. 37 1.3.1 The Tetrahedron....................................... 37 1.3.2 The Cube........................................... 43 1.3.3 The Dodecahedron...................................... 47 1.4 Group Actions............................................ 52 2 The Orbit-Stabilizer Theorem 59 2.1 Matrices................................................ 59 2.2 Necklaces............................................... 65 2.3 Platonic Solids............................................ 79 2.3.1 The Cube........................................... 79 2.3.2 The Tetrahedron....................................... 93 2.3.3 The Octahedron....................................... 101 2.3.4 The Dodecahedron...................................... 107 2.3.5 Additional Comments.................................... 109 2.4 Orbit Stabilizer Questions on Graphs................................ 111 2.4.1 Graphs and Labeled Graphs on Three Vertices...................... 111 2.4.2 Graphs and Labeled Graphs on Four Vertices....................... 113 2.5 The Proof of the Orbit Stabilizer Theorem............................. 119 3 4 CONTENTS 3 The Cauchy-Frobenius Lemma 125 3.1 Necklaces............................................... 125 3.2 Graphs................................................. 142 3.3 Platonic Solids............................................ 147 3.4 Matrices................................................ 168 3.5 The Proof of the Cauchy Frobenius Theorem........................... 176 4 P´olya Enumeration Questions 181 4.1 Necklaces............................................... 181 4.2 Graphs................................................. 187 4.3 Platonic Solids............................................ 191 4.4 Matrices................................................ 194 5 Irreducible Polynomials Over Finite Fields 201 5.1 Finding Irreducible Polynomials Over Z2 .............................. 202 5.2 Finding Irreducible Polynomials Over Z3 .............................. 206 5.3 Finding Irreducible Polynomials of Over Z5 ............................ 210 5.4 Finding Irreducible Polynomials of Over Z7 ............................ 215 5.5 Finding Irreducible Polynomials of Over Z11 ............................ 220 6 Construction of Finite Fields 223 6.1 Finite Fields of Order 2n....................................... 223 6.2 Finite Fields of Order 3n....................................... 226 6.3 Finite Fields of Order 5n....................................... 231 6.4 Finite Fields of Order 7n....................................... 235 6.5 Finite Fields of Order 11n....................................... 237 7 Linear Codes 243 7.1 Linear [n; k; d]-Codes and Duals................................... 243 7.2 Hadamard Codes........................................... 247 7.3 Automorphism Groups of Codes.................................. 252 8 Acknowledgments 257 Chapter 1 Some Basic Results in Group Theory 1.1 The Dihedral Groups There are many different ways one can define the dihedral groups. It's helpful to first look at them as actual reflections and rotations of some object. We can define these groups by flipping and rotating this object and then making a multiplication table out of the results. This becomes impractical for large values of n, but works very well up to about D5. For an example, we can start by taking a college textbook, or to be precise, a two-dimensional image of one. Picture it lying face up in some plane, and picture also a line straight through the middle of the book. There are many choices for such a line, and it actually doesn't matter which we choose, but it's probably easiest to imagine one parallel to two of the edges of the book. We start by defining two group elements. The element f in Dn corresponds to the motion of flipping the book over that axis, so it is now face down with the top of the book pointing the opposite direction as before. The element r corresponds to rotating the book clockwise 360=n degrees through a point directly in its center. For example, r2f is the motion we get from flipping and then rotating twice, and frfr2 is the motion we get from rotating twice, flipping, rotating, and then flipping again. We perform these operations from right to left in all cases in order to match the common notation for compositions used in Pre-Calculus and Calculus, where f ◦ g equals f(g(x)) and g ends up applied first. We have an equivalence relation on the set of all the possible motions that arise from combining these operations, considering two of these motions to be equal if they affect our object in the same way. This set of equivalence classes under composition forms the dihedral group. It helps to examine the relationship between f and r that arises because we set two of these motions to be equivalent if they leave the book in the same position. For example, in D4, if we rotate and flip, the book is in the same position as if we had flipped first and rotated three times, and thus fr = r3f: This is illustrated in figure 1.1 where we label a square book with the numbers one through four, and illustrate fr, rf and then r3f. The dotted line depicts the axis for reflection, which does not change as the book is rotated. Every 2 n−1 2 n−1 motion in Dn ends up equivalent to one of the following: e; r; r ; ··· ; r ; f; rf; r f; ··· ; r f and thus Dn contains 2n elements. It is important to keep in mind that the axis over which we flip the book does not rotate together with the object we are rotating. No matter which position the book is in, when we apply the flip, we always flip over the initial axis. It is also important to note that the elements in the group are the actual motions, and not the book itself. We use the positions to determine if two motions are equivalent, by seeing if the leave 5 6 CHAPTER 1. SOME BASIC RESULTS IN GROUP THEORY 3 Figure 1.1: Showing that fr 6= rf and fr = r f in D4 the book in the same place. We understand that this method of first defining the dihedral groups is not particularly rigorous. For 501 499 example, in D1000 it would be particularly difficult to see that fr is equal to r f: However, for small enough values of n one can get through most of the questions simply by taking a book and making these flips, and we feel it is nice to introduce these groups through an actual process that students can put their hands on. Another way to construct Dn is by looking at it as a subgroup of the symmetric group Sn. For even n this is the smallest group containing the permutation (1; 2; 3; ··· ; n) which represents r and (1; n)(2; n − 1) ··· (n=2; (n=2) + 1) which represents f. This will sometimes be useful when the numbers correspond to the physical corners and edges of some object. If we recall how to multiply in the symmetric group we can find 3 the relations for our group by simply taking products. For example we can see that rf = fr in D4 because (1; 2; 3; 4)(1; 4)(2; 3) and (1; 4)(2; 3)(1; 2; 3; 4)3 both equal (2; 4). There is an alternate construction for odd n where r is still (1; 2; 3; ··· ; n) but f is now (1; n)(2; n − 1) ··· ((n − 1)=2; (n + 1)=2): We will look into this construction more in the section on symmetric groups. n 2 For these problems it is enough to consider the group presented as Dn =< r; fjr = f = e; fr = rn−1f >. There is more than one form for this presentation, Note that there are some prettier ways to do this, but this form has some distinct advantages. Every element can be written in the form rif j where i 2 f0; 1; 2; ··· ; (n − 1)g and j 2 f0; 1g. Then we can quickly simplify any product simply by pushing every r to the right of an f past that f, turning it into a rn−1. 1.1.1 Arbitrary Dihedral Group Questions 1. Use the fact that fr = rn−1f to prove that frk = rn−kf. [Answer: Moving each r past our f shows frk = (rn−1)kf = rnk−kf = rn(k−1)+n−k = (rn)k−1rn−k = ek−1rn−k = rn−k: Note that this is equivalent to the statement (n − 1)k ≡ −k modulo n.] k 2. Prove that for any k ≥ 0, an element of the form r f is its own inverse in Dn. [Answer: We just need to show the square of any such element is the identity. (rnf)2 = (rnf)(rnf) = 1.1. THE DIHEDRAL GROUPS 7 2 2 rn(rn−1)nff = rn(rn(n−1))e = r(n+n −n) = r(n ) = (rn)n = en = e:] 3. Prove that for odd n, Dn has n elements of order two. [Answer: We have already shown that rkf has order two for 0 ≤ k < n. This gives us n elements of order two. We must show there are no more. The identity always has order one. We have to show that no element from r; r2; ··· ; rn−1 is its own inverses. Take any rk for 0 < k < n. Then (rk)2 = r2k. If this is the identity then n would have to divide 2k. As n is odd, it would also have to divide k. This cannot happen with 0 < k < n.] 4. Prove that for even n, Dn has n + 1 elements of order two. [Answer: We have already shown that rkf has order two for 0 ≤ k < n. The identity has order one. We must show that only one element from r; r2; ··· ; rn−1 has order two.