Let be a group. Assume that is a partition of such that is a group under the “quotient” operation as follows: to compute the product of and take any representatives and and let be the element of containing . (Assume this operation is well defined.)
Prove that the element of containing 1 is a normal subgroup of and that the elements of are the cosets of , so that .
We will denote the equivalence class of by ; hence .
Lemma 1: Let and be as described above. Then for all , . Proof: we have . We have , so that .
Lemma 2: Let and be as described above. Then for all , . Proof: We have . We have , so that .
First we show that . Note that is nonempty since . Now suppose . We have , and so by the uniqueness of group inverses, . Thus , and so . By the Subgroup Criterion, is a subgroup of .
Now we show that is normal in . Let ; now by Lemma 1.
Now we show that every is a coset of . Let and . Now , using Lemma 2. So is a coset of .
Now we show that every coset of is in . Let . Since is a partition, for some . Thus , using Lemma 2, and so .