Equivalent Statements for Congruence Modulo Subgroup/Left

Theorem

Let $G$ be a group.

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

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

Then the following statements are equivalent:

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

Proof

 $\displaystyle x$ $\equiv^l$ $\displaystyle y \pmod H$ $\displaystyle \iff \ \$ $\displaystyle x^{-1} y$ $\in$ $\displaystyle H$ Definition of Left Congruence Modulo Subgroup $\displaystyle \iff \ \$ $\displaystyle \exists h \in H: x^{-1} y$ $=$ $\displaystyle h$ Definition of Element of $H$ $\displaystyle \iff \ \$ $\displaystyle \exists h \in H: y$ $=$ $\displaystyle x h$ Division Laws for Groups

$\blacksquare$