# 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 $y \in xH$.

### Proving that left cosets give a left congruence

This is more or less direct. PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]