# Element in Left Coset iff Product with Inverse in Subgroup

## Theorem

Let $G$ be a group.

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

Let $x, y \in G$.

Let $y H$ denote the left coset of $H$ by $y$.

Then:

$x \in y H \iff x^{-1} y \in H$

## Proof

 $\displaystyle x$ $\in$ $\displaystyle y H$ $\displaystyle \iff \ \$ $\displaystyle \exists h \in H: \ \$ $\displaystyle x$ $=$ $\displaystyle y h$ Definition of Left Coset $\displaystyle \iff \ \$ $\displaystyle \exists h \in H: \ \$ $\displaystyle x^{-1}$ $=$ $\displaystyle h^{-1} y^{-1}$ Inverse of Group Product $\displaystyle \iff \ \$ $\displaystyle \exists h \in H: \ \$ $\displaystyle x^{-1} y$ $=$ $\displaystyle h^{-1}$ Product with $y$ on the right $\displaystyle \iff \ \$ $\displaystyle x^{-1} y$ $\in$ $\displaystyle H$ $H$ is a subgroup

$\blacksquare$