Concept:

Left coset:

Let G be a group and H be a subgroup of G. Let a be an element of G. Then the subset {ah: h ∈ H} is called a left coset of H in G and is denoted by aH.

Right coset:

Let G be a group and H be a subgroup of G. Let a be an element of G. Then the subset {ha: h ∈ H} is called a right coset of H in G and is denoted by Ha.

Normal Subgroup:

Let G be a group, H be a subgroup of G. Then H is said to be a normal subgroup of G, if x h x-1 ∈ H, ∀ x ∈ G and ∀ h ∈ H.

Note:

• For a normal subgroup left coset is equal to right coset.
• If G is an abelian group then every subgroup of G is a normal subgroup. 

Calculation:

Given: G = (Z, +) and H = (4Z, +) is a subgroup of G.

G = (Z, +) is an abelian group

As we know that, if G is an abelian group then every subgroup of G is a normal subgroup.

∴ H is a normal subgroup

So, all the left and right cosets of H are the same.

Now, let's find out the cosets of H in G.

The distinct cosets of H in G are:

0 + H = {4n : n ∈ Z} = H

1 + H = {4n + 1: n ∈ Z}

2 + H = {4n + 2: n ∈ Z}

3 + H = {4n + 3: n ∈ Z}

So, H, 1 + H, 2 + H, 3 + H are the cosets of H in G.

Hence, there are 4 cosets of H in G