Left congruence

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This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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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.


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