# Equivalent Statements for Congruence Modulo Subgroup/Right

## Theorem

Let $G$ be a group.

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

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

Then the following statements are equivalent:

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

## Proof

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

$\blacksquare$