Right congruence

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

Definition

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 $y\in Hx$ .