Equivalent Statements for Congruence Modulo Subgroup

Theorem

Let $G$ be a group.

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

Left Congruence

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

Then the following statements are equivalent:

 $(1):\quad$ $\displaystyle x$ $\equiv^l$ $\displaystyle y \pmod H$ $(2):\quad$ $\displaystyle x^{-1} y$ $\in$ $\displaystyle H$ $(3):\quad$ $\displaystyle \exists h \in H: x^{-1} y$ $=$ $\displaystyle h$ $(4):\quad$ $\displaystyle \exists h \in H: y$ $=$ $\displaystyle x h$

Right Congruence

Let $x \equiv^r y \pmod H$ denote that $x$ is right congruent modulo $H$ to $y$.

Then the following statements are equivalent:

 $(1):\quad$ $\displaystyle x$ $\equiv^r$ $\displaystyle y \pmod H$ $(2):\quad$ $\displaystyle x y^{-1}$ $\in$ $\displaystyle H$ $(3):\quad$ $\displaystyle \exists h \in H: x y^{-1}$ $=$ $\displaystyle h$ $(4):\quad$ $\displaystyle \exists h \in H: x$ $=$ $\displaystyle h y$