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For example, the permutation  à 1 2 3 4 5 6 7 D 4 3 1 2 6 5 7 in S7 is identified with the bijective map of f1; 2; : : : ; 7g that sends 1 to 4, 2, to 3, 3 to 1, and so on. It should be clear, upon reflection, that the multiplication of permutations is the same as the composition of bijective maps. Thus the three properties listed for permutations follow immediately from the corresponding properties of bijective maps. f1; 2; : : : ; ng/. 1/. In fact, the image of 1 under an invertible map can be any of the n numbers 1; 2; : : : ; n; for each of these possibilities, there are n 1 possible images for 2, and so forth.

A C b/ a0 and b b 0 are divisible by n. b a0 b 0 / b0/ is divisible by n. ■ We denote by Zn the set of residue classes modulo n. The set Zn has a natural algebraic structure which we now describe. Let A and B be elements of Zn , and let a 2 A and b 2 B; we say that a is a representative of the residue class A, and b a representative of the residue class B. The class Œa C b and the class Œab are independent of the choice of representatives. 5. Thus Œa C b D Œa0 C b 0  and Œab D Œa0 b 0 . This means that it makes sense to define A C B D Œa C b and AB D Œab.

The product of non-zero integers is non-zero. We write a > b if a b > 0 and a b if a b 0. Then > is a total order on the integers. That is, it is never true that a > a; for distinct integers a; b, either a > b or b > a; and whenever a > b and b > c, it follows that a > c. 1 is that jabj maxfjaj; jbjg for non-zero integers a and b. ✐ ✐ ✐ ✐ ✐ ✐ “bookmt” — 2006/8/8 — 12:58 — page 26 — #38 ✐ 26 ✐ 1. ALGEBRAIC THEMES Although the multiplicative structure of the integers is subordinate to the additive structure, many of the most interesting properties of the integers have to do with divisibility, factorization, and prime numbers.

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