Q:

The equivalence class of the identity element of $G$ defined by $a \sim b$ iff $ab^{-1} \in H$ for $H\le G$.

I already proved that $a \sim b$ is an equivalence relationship. 
I think that identity element $e$ of $G$ is defined as: $b b^{-1}$ because $(ab^{-1})(bb^{-1}) = ab^{-1}$
Now I'm just realizing that I also have to prove that $e\ a = a$. 
so $(aa^{-1})(ab^{-1}) = ab^{-1}$ as well. 
so either $a a^{-1}$ or $b b^{-1}$ are both identity elements? 
I'm also getting caught up on what an equivalence class for an identity element actually is. 

A:

Since $a \sim e \Leftrightarrow ae^{-1}=a \in H$, the equivalence class of $e$ is $H$. More generally, the equivalence class of $b\in G$ is the right coset $Hb$.

