Left congruence

This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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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:

$a\equiv b⟹ca\equiv cb$

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 $x\in G$, $x\equiv xh$ (pre-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

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