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 is an equivalence relation on such that:
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 , so is : The proof of this comes as follows. Suppose . Then . We already know that . Hence, by the transitivity of , we have .
- Closure under inverses: If , then we can pre-multiply both sides by and obtain
Let denote this subgroup. Then clearly, for any , (left multiplying by ). Thus all the elements in the left coset of are in the same equivalence class as .
Further, we can show that if , they must be in the same left coset. Suppose . Then, left multpily both sides by . This gives , hence or .