I'm trying to solve the problem 2.10.3 from Artin's Algebra first edition:
Let P be a partition of a group G with the property that for any pair of elements A, B of the partition, the product set AB is contained entirely within another element C of the partition. Let N be the element of P which contains 1. Prove that N is a normal subgroup of G and that P is the set of its cosets.
I manage to prove that N is a group, that is normal, and that is't cosets are contained in the elements of the partition but I don't see how to prove the converse inclusion, any hint?