Definition (Coset)

Given a group $G$ and a subgroup $H$,we define an equivalence relation $\sim$ where $$a\sim b\iff ab^{-1}\in H$$this relation partitions $G$ into equivalence classes $aH$ for some $a\in G$. These are called cosets of $H$.

Definition (Index)

Given a group $G$ and subgroup $H$ where we have defined for some $a\in G$ the cosets of $H$. Their index is defined as the number of cosets and is denoted as $$[G:H]$$