Abstract Algebra Theory and Applications - Computer Science ...

36 CHAPTER 2 GROUPSZ n . Consider the integers modulo 12 and the corresponding partition of theintegers:[0] = {. . . , −12, 0, 12, 24, . . .},[1] = {. . . , −11, 1, 13, 25, . . .},.[11] = {. . . , −1, 11, 23, 35, . . .}.When no confusion can arise, we will use 0, 1, . . . , 11 to indicate the equivalenceclasses [0], [1], . . . , [11] respectively. We can do arithmetic on Z n . Fortwo 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 modulon 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) 7 · 3 ≡ 1 (mod 5)3 + 5 ≡ 0 (mod 8) 3 · 5 ≡ 7 (mod 8)3 + 4 ≡ 7 (mod 12) 3 · 4 ≡ 0 (mod 12).In particular, notice that it is possible that the product of two nonzeronumbers modulo n can be equivalent to 0 modulo n.Table 2.1. Multiplication table for Z 8· 0 1 2 3 4 5 6 70 0 0 0 0 0 0 0 01 0 1 2 3 4 5 6 72 0 2 4 6 0 2 4 63 0 3 6 1 4 7 2 54 0 4 0 4 0 4 0 45 0 5 2 7 4 1 6 36 0 6 4 2 0 6 4 27 0 7 6 5 4 3 2 1Example 2. Most, but not all, of the usual laws of arithmetic hold foraddition and multiplication in Z n . For instance, it is not necessarily truethat there is a multiplicative inverse. Consider the multiplication table forZ 8 in Table 2.1. Notice that 2, 4, and 6 do not have multiplicative inverses;that is, for n = 2, 4, or 6, there is no integer k such that kn ≡ 1 (mod 8).