By Thomas W. Judson

This article covers the conventional strategy of teams, earrings, fields with the mixing of computing and purposes present in parts resembling coding thought and cryptography. utilized examples are used to help within the motivation of studying to end up theorems and propositions. the character of routines during this textual content variety over a number of different types together with computational, conceptual and theoretical. those workouts and difficulties enable the exploration of recent effects and concept. The versatile association can be utilized in lots of other ways to stress conception or functions. It comprises beneficial properties and in textual content studying aids, functions inside of each bankruptcy, volume and caliber of examples and workouts, supplementary themes, stability of conception and arithmetic, old notes, and computing device technology tasks.

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Cayley table for (Z5 , +) + 0 1 2 3 4 0 0 1 2 3 4 1 1 2 3 4 0 2 2 3 4 0 1 3 3 4 0 1 2 4 4 0 1 2 3 It is often convenient to describe a group in terms of an addition or multiplication table. Such a table is called a Cayley table. Example 4. The integers mod n form a group under addition modulo n. Consider Z5 , consisting of the equivalence classes of the integers 0, 1, 2, 3, 42 CHAPTER 2 GROUPS and 4. We define the group operation on Z5 by modular addition. We write the binary operation on the group additively; that is, we write m + n.

Suppose that p does not divide a. We must show that p | b. Since gcd(a, p) = 1, there exist integers r and s such that ar + ps = 1. So b = b(ar + ps) = (ab)r + p(bs). Since p divides both ab and itself, p must divide b = (ab)r + p(bs). 7 (Euclid) There exist an infinite number of primes. Proof. We will prove this theorem by contradiction. Suppose that there are only a finite number of primes, say p1 , p2 , . . , pn . Let p = p1 p2 · · · pn + 1. We will show that p must be a different prime number, which contradicts the assumption that we have only n primes.

Compute the subgroups of the symmetry group of a square. 35. Let H = {2k : k ∈ Z}. Show that H is a subgroup of Q∗ . 36. Let n = 0, 1, 2, . . and nZ = {nk : k ∈ Z}. Prove that nZ is a subgroup of Z. Show that these subgroups are the only subgroups of Z. 37. Let T = {z ∈ C∗ : |z| = 1}. Prove that T is a subgroup of C∗ . 38. Let G consist of the 2 × 2 matrices of the form cos θ sin θ − sin θ cos θ where θ ∈ R. Prove that G is a subgroup of SL2 (R). 39. Prove that √ G = {a + b 2 : a, b ∈ Q and a and b are not both zero} is a subgroup of R∗ under the group operation of multiplication.