In previous chapters we looked at some of the uses of the multiplicative group of the nth roots of unity. Under the name of the cyclic group of order n, it furnishes the simplest example of an Abelian group whilst under the name of the additive group of the integers modulo n, it forms a simple but very useful tool in number theory. In this chapter we introduce another kind of Abelian group which also plays an important role in number theory.

Let ℤn be the set of integers mod n. (More formally ℤn is the collection of equivalence classes [a] = {b:b ≡ a mod n}.) Recall that b and c are said to be coprirne if they have no common factors. We make an obvious remark.

Lemma 101.1. (i) if a and n are coprirne so are a + nm and n.

Proof. If q divides n and a + nm then q divides n and a = a + nm – nm.

Thus we can define

G = {[a]∈ℤn:a and n are coprirne}

without ambiguity.

In the same vein we have the following results.

Lemma 101.1. (ii) If [a1] = [a2] and [b1] = [b2] then [a1b1] = [a2b2].

(iii) If [a], [b]∈G then [ab] ∈ G.

Proof. (ii) Since a1 ™ a2, b1 ™ b2 ≡ 0 mod n we have

a1b1 – a2b2 = (a1 – a2)b1 + a2(b1 – b2) ≡ 0 mod n,

as required.