Let be a group, a normal subgroup of and the set of cosets of Then is a partition of and for all We’d like to consider the converse of this.

**Problem**. Let be a group and suppose that is a set partition of which satisfies the following condition:

for every there exists such that

Let be the element of which contains the identity element of Prove that is a normal subgroup of and is the set of cosets of in

**Solution**. Notice that an obvious result of is that if and then for some Now, if then and so We also have for some by Thus and so Therefore i.e. is multiplicatively closed. Let Then for some and since we get Thus and so Hence and so is a subgroup. To prove that is normal, let Then, by there exists such that But then because and so Thus i.e. is normal. Finally, let and choose such that and Then because and so Hence Let be such that Then because and so Hence