Left Congruence Class Modulo Subgroup is Left Coset

Theorem

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

Let $\RR^l_H$ be the equivalence defined as left congruence modulo $H$.

The equivalence class $\eqclass g {\RR^l_H}$ of an element $g \in G$ is the left coset $g H$.

This is known as the left congruence class of $g \bmod H$.

Proof

Let $x \in \eqclass g {\RR^l_H}$.

Then:

$\ds x$

$\in$

$\ds \eqclass g {\RR^l_H}$

$\ds \leadsto \ \ $

$\ds \exists h \in H: \, $

$\ds g^{-1} x$

$=$

$\ds h$

Definition of Left Congruence Modulo $H$

$\ds \leadsto \ \ $

$\ds \exists h \in H: \, $

$\ds x$

$=$

$\ds g h$

Group properties

$\ds \leadsto \ \ $

$\ds x$

$\in$

$\ds g H$

Definition of Left Coset

$\ds \leadsto \ \ $

$\ds \eqclass g {\RR^l_H}$

$\subseteq$

$\ds g H$

Definition of Subset

Now let $x \in g H$.

Then:

$\ds x$

$\in$

$\ds g H$

$\ds \leadsto \ \ $

$\ds \exists h \in H: \, $

$\ds x$

$=$

$\ds g h$

Definition of Left Coset

$\ds \leadsto \ \ $

$\ds g^{-1} x$

$=$

$\ds h \in H$

Group properties

$\ds \leadsto \ \ $

$\ds x$

$\in$

$\ds \eqclass g {\RR^l_H}$

Definition of Left Congruence Modulo $H$

$\ds \leadsto \ \ $

$\ds $

$=$

$\ds g H \subseteq \eqclass g {\RR^l_H}$

Definition of Subset

Thus:

$\eqclass g {\RR^l_H} = g H$

that is, the equivalence class $\eqclass g {\RR^l_H}$ of an element $g \in G$ equals the left coset $g H$.

$\blacksquare$

Also see

Right Congruence Class Modulo Subgroup is Right Coset

Sources

1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures: Theorem $11.1$
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): $\S 42.3$ Another approach to cosets
1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $5$: Cosets and Lagrange's Theorem: Proposition $5.4$