## Equivalent characterization of group congruences

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 .

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## Comments

I like your notations. A typo: should be .

Fixed. Thanks!