Let \(G\) be a group and \(H\leq G\) a subgroup. The index of \(H\) in \(G\), written \([G:H]\), is the cardinality of the set \(G/H=\theset{gH \st g\in G}\) of left cosets of \(H\).
The map \(gH\mapsto Hg^{-1}\) is a well-defined bijection from left cosets to right cosets, since \(g_1H=g_2H\) if and only if \(g_2^{-1}g_1\in H\) if and only if \(Hg_1^{-1}=Hg_2^{-1}\). So \([G:H]\) is also the number of right cosets of \(H\).