# Left Coset Equals Subgroup iff Element in Subgroup

## Theorem

Let $G$ be a group whose identity is $e$.

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

Let $x \in G$.

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

Then:

$x H = H \iff x \in H$

## Proof

 $\ds x H$ $=$ $\ds H$ $\ds \leadstoandfrom \ \$ $\ds x H$ $=$ $\ds e H$ Left Coset by Identity: $e H = H$ $\ds \leadstoandfrom \ \$ $\ds x e^{-1}$ $\in$ $\ds H$ Left Cosets are Equal iff Product with Inverse in Subgroup $\ds \leadstoandfrom \ \$ $\ds x$ $\in$ $\ds H$ Group Properties

$\blacksquare$