When a subgroup of a group G is given, the subgroup provides a way of partitioning the group into disjoint sets, all of the same size. This partition is called a partition into (left or right) cosets of the subgroup. The fact that the cosets are all of the same size enables us to show that the number of elements in a subgroup must be a factor of the number of elements in the whole group when the group contains a finite number of elements in all.

Concurrent reading: Birkhoff and MacLane, chapter 6, section on Lagrange's theorem; Green, chapter 6, sections 1–5; Fraleigh, section 11.

1. 1 A relation R is defined on the group (ℤ, +) by a R b when a - b ∈ 〈3〉. Prove that R is an equivalence relation and find the equivalence classes. The three classes are called the residue classes modulo 3 and a R b in this case is usually written a ≡ b (mod 3).

2. 2 A relation R is defined on the group (ℤ, +) by a R b when a - b ∈ 〈n〉. Prove that R is an equivalence relation and find the equivalence classes. The n classes are called the residue classes modulo n and a R b in this case is usually written a ≡ b (mod n).

3. 3 If the group C6 is generated by the element a and a relation R is defined on C6 by

4. x R y when xy-1 ∈ 〈a3〉,

5. prove that R is an equivalence relation and find the equivalence classes.

6. 3 […]