Right congruence

''The notion of right congruence also makes sense in the more general context of a monoid. In fact, the same definition works.''

Symbol-free definition
A right congruence on a group is an equivalence relation on the group with the property that the equivalence relation is preserved on right multiplication by any element of the group.

Definition with symbols
A right congruence on a group $$G$$ is an equivalence relation $$\equiv$$ on $$G$$ such that:

$$a \equiv b \implies ac \equiv bc$$

Relation with other notions
The dual notion to right congruence is the notion of left congruence.

An equivalence relation is termed a congruence if it is both a left congruence and a right congruence.

Correspondence between subgroups and right congruences
The following is true:

Right congruences are precisely the equivalence relations whose equivalence classes are the right cosets of a subgroup

Proving that any right congruence gives right cosets
We first show that the equivalence class of the identity element is a subgroup. For this, we show the following three things:


 * Identity elements:The identity element is equivalent to the identity element: This follows on account of the relation being reflexive
 * Closure under multiplication: If $$a,b \equiv e$$, so is $$ab$$: The proof of this comes as follows. Suppose $$a \equiv e$$. Then $$ab \equiv b$$. We already know that $$b \equiv e$$. Hence, by the transitivity of $$\equiv$$, we have $$ab \equiv e$$.
 * Closure under inverses: If $$a \equiv e$$, then we can right multiply both sides by $$a^{-1}$$ and obtain $$e \equiv a^{-1}$$

Let $$H$$ denote this subgroup. Then clearly, for any $$x \in G$$, $$x \equiv hx$$ (right multiplying $$e \equiv h$$ by $$x$$). Thus all the elements in the right coset of $$H$$ are in the same equivalence class as $$x$$.

Further, we can show that if $$x \equiv y$$, they must be in the same right coset. Suppose $$x \equiv y$$. Then, right multiply both sides by $$y^{-1}$$. This gives $$xy^{-1} \equiv e$$, hence $$xy^{-1} \in H$$ or $$y \in Hx$$.

Proving that right cosets give a right congruence
This is more or less direct.