Group theory/Cosets/Properties/Fact

Let ${}G$ be a group, and let ${}H\subseteq G$ be a subgroup. Let ${}x,y\in G$ be elements.

Then the following statements are equivalent.

1. ${}x\in yH$.
2. ${}y\in xH$.
3. ${}y^{-1}x\in H$.
4. ${}x^{-1}y\in H$.
5. ${}xH\cap yH\neq \emptyset$.
6. ${}x\sim _{H}y$.
7. ${}xH=yH$.

Proof, Write another proof