Left congruence

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

Definition

Symbol-free definition

A left congruence on a group is an equivalence relation on the group with the property that the equivalence relation is preserved on left multiplication by any element of the group.

Definition with symbols

A left congruence on a group $G$ is an equivalence relation $\equiv$ on $G$ such that:

$a\equiv b\implies ca\equiv cb$

Relation with other notions

The dual notion to left congruence is the notion of right congruence.

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

Correspondence between subgroups and left congruences

The following is true:

Left congruences are precisely the equivalence relations whose equivalence classes are the left cosets of a subgroup

Proving that any left congruence gives left 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 $b\equiv e$ . Then $ab\equiv a$ . We already know that $a\equiv e$ . Hence, by the transitivity of $\equiv$ , we have $ab\equiv e$ .
Closure under inverses: If $a\equiv e$ , then we can pre-multiply both sides by $a^{-1}$ and obtain $e\equiv a^{-1}$

Let $H$ denote this subgroup. Then clearly, for any $x\in G$ , $x\equiv xh$ (left multiplying $e\equiv h$ by $x$ ). Thus all the elements in the left 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 left coset. Suppose $x\equiv y$ . Then, left multpily both sides by $y^{-1}$ . This gives $y^{-1}x\equiv e$ , hence $y^{-1}x\in H$ or $y\in xH$ .