Left Congruence Modulo Subgroup is Equivalence Relation

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Theorem

Let $G$ be a group, and let $H$ be a subgroup of $G$.

Let $x, y \in G$.

Let $x \equiv^l y \pmod H$ denote the relation that $x$ is left congruent modulo $H$ to $y$.

Then the relation $\equiv^l$ is an equivalence relation.

Proof 1

Let $G$ be a group whose identity is $e$.

Let $H$ be a subgroup of $G$.

For clarity of expression, we will use the notation:

$\tuple {x, y} \in \RR^l_H$

for:

$x \equiv^l y \pmod H$

From the definition of left congruence modulo a subgroup, we have:

$\RR^l_H = \set {\tuple {x, y} \in G \times G: x^{-1} y \in H}$

We show that $\RR^l_H$ is an equivalence:

Reflexive

We have that $H$ is a subgroup of $G$.

From Identity of Subgroup:

$e \in H$

Hence:

$\forall x \in G: x^{-1} x = e \in H \implies \tuple {x, x} \in \RR^l_H$

and so $\RR^l_H$ is reflexive.

$\Box$

Symmetric

$\ds \tuple {x, y}$

$\in$

$\ds \RR^l_H$

$\ds \leadsto \ \ $

$\ds x^{-1} y$

$\in$

$\ds H$

Definition of Left Congruence Modulo $H$

$\ds \leadsto \ \ $

$\ds \tuple {x^{-1} y}^{-1}$

$\in$

$\ds H$

Group Axiom $\text G 0$: Closure

But then:

$\tuple {x^{-1} y}^{-1} = y^{-1} x \implies \tuple {y, x} \in \RR^l_H$

Thus $\RR^l_H$ is symmetric.

$\Box$

Transitive

$\ds \tuple {x, y}, \tuple {y, z}$

$\in$

$\ds \RR^l_H$

$\ds \leadsto \ \ $

$\ds x^{-1} y$

$\in$

$\ds H$

Definition of Left Congruence Modulo $H$

$\, \ds \land \, $

$\ds y^{-1} z$

$\in$

$\ds H$

Definition of Left Congruence Modulo $H$

$\ds \leadsto \ \ $

$\ds \tuple {x^{-1} y} \tuple {y^{-1} z} = x^{-1} z$

$\in$

$\ds H$

Group Properties

$\ds \leadsto \ \ $

$\ds \tuple {x, z}$

$\in$

$\ds R^l_H$

Definition of Left Congruence Modulo $H$

Thus $\RR^l_H$ is transitive.

$\Box$

So $\RR^l_H$ is an equivalence relation.

$\blacksquare$

Proof 2

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Let $G$ be a group, and let $H$ be a subgroup of $G$.

Then $G$ is a fortiori a monoid.

From Condition for Cosets of Subgroup of Monoid to be Partition, $\RR^l_H$ is a relation induced by partition.

The result follows from Relation Induced by Partition is Equivalence.

$\blacksquare$

Also see

Definition:Left Coset
Definition:Left Coset Space

Right Congruence Modulo Subgroup is Equivalence Relation

Sources

1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 6.1$. The quotient sets of a subgroup: Lemma $\text {(ii)}$
1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures: Theorem $11.1$
1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.9$: Subgroups: Theorem $11$
1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 37$
1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 42.1$ Another approach to cosets
1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $5$: Cosets and Lagrange's Theorem: Proposition $5.3$