By Thomas W. Judson

ISBN-10: 0534936849

ISBN-13: 9780534936846

This article covers the normal process of teams, jewelry, fields with the mixing of computing and functions present in components reminiscent of coding conception and cryptography. utilized examples are used to help within the motivation of studying to turn out 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 permit the exploration of recent effects and conception. The versatile association can be utilized in lots of other ways to stress concept or purposes. It comprises positive factors and in textual content studying aids, functions inside each bankruptcy, volume and caliber of examples and workouts, supplementary subject matters, stability of idea and arithmetic, historic notes, and desktop technological know-how tasks.

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3. If h ∈ H, then h−1 ∈ H. Proof. First suppose that H is a subgroup of G. We must show that the three conditions hold. Since H is a group, it must have an identity eH . We must show that eH = e, where e is the identity of G. We know that EXERCISES 49 eH eH = eH and that eeH = eH e = eH ; hence, eeH = eH eH . By right-hand cancellation, e = eH . The second condition holds since a subgroup H is a group. To prove the third condition, let h ∈ H. Since H is a group, there is an element h ∈ H such that hh = h h = e.

11] respectively. We can do arithmetic on Zn . For two integers a and b, define addition modulo n to be (a+b) (mod n); that is, the remainder when a + b is divided by n. Similarly, multiplication modulo n is defined as (ab) (mod n), the remainder when ab is divided by n. Example 1. The following examples illustrate integer arithmetic modulo n: 7 + 4 ≡ 1 (mod 5) 3 + 5 ≡ 0 (mod 8) 3 + 4 ≡ 7 (mod 12) 7 · 3 ≡ 1 (mod 5) 3 · 5 ≡ 7 (mod 8) 3 · 4 ≡ 0 (mod 12). In particular, notice that it is possible that the product of two nonzero numbers modulo n can be equivalent to 0 modulo n.

Let G be a group and suppose that (ab)2 = a2 b2 for all a and b in G. Prove that G is an abelian group. 52 CHAPTER 2 GROUPS 32. Find all the subgroups of Z3 × Z3 . Use this information to show that Z3 × Z3 is not the same group as Z9 . 33. Find all the subgroups of the symmetry group of an equilateral triangle. 34. 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.

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