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
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.
$\blacksquare$
Also see
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 $5.3$