Comparing cosets

Suppose $G$ is a group, $H \leq G$ a subgroup, and $a, b \in G$ . Prove that the following are equivalent:

$a H = b H$
$b \in a H$
$b^{- 1} a \in H$

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Suppose $G$ is a group, $H\leq G$ a subgroup, and $a,b\in G$. Prove that the following are equivalent:
\begin{enumerate}[label=\alph*)]
	\item $aH=bH$
	\item $b\in aH$
	\item $b^{-1}a\in H$
\end{enumerate}