Congruence Class Modulo Subgroup is Coset

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 $g \bmod H$.

Also see

• Coset Space forms Partition
• Element of Group is in Unique Coset of Subgroup