After proving that cosets partition a group, we now need efficient tests for deciding whether two elements determine the same coset. Instead of listing all elements, we can use a product involving inverses. For left cosets, the key expression is $a^{-1}\circ b$. For right cosets, the key expression is $b\circ a^{-1}$. These criteria are practical tools: they turn questions about subsets into questions about whether a single element lies in the subgroup.

Let $(G,\circ)$ be a group and let $H$ be a subgroup of $G$. If $a,b\in G$, then $$\begin{equation*} b\in aH \iff a^{-1}\circ b\in H. \end{equation*}$$