Correspondence between normal subgroups and congruence relations

A Normal subgroup uniquely defines a congruence relation on and vice versa, #m/thm/group such that , and for any the congruence classes correspond to cosets of :

Proof

First, we will prove that is a subgroup. Clearly . Let , i.e. . Then and therefore . Therefore is a subgroup by One step subgroup test.

Next, we will show that is a Normal subgroup. Let and . Then and thus . Hence for any , Therefore is normal.

Finally we show the equivalence between (left) cosets of and congruence classes. For any

as required.

As a result of this theorem, normal subgroups may be used to form a Quotient group (following the usual notion of Algebraic quotient) where each coset is taken as a group element.

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