Congruence Class Modulo Subgroup is Coset
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Theorem
Let $G$ be a group.
Let $H$ be a subgroup of $G$.
Left Congruence Class
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 $g \bmod H$.
Right Congruence Class
Let $\RR^r_H$ be the equivalence defined as right congruence modulo $H$.
The equivalence class $\eqclass g {\RR^r_H}$ of an element $g \in G$ is the right coset $H g$.
This is known as the right congruence class of $g \bmod H$.