Definition
For group , and subgroup , we define the set of all left cosets in with respect to as the quotient .
We have that the number of left cosets in is equal to or the index of in .
For group G, and subgroup H≤G, we define the set of all left cosets in G with respect to H as the quotient G/H .
We have that the number of left cosets in G is equal to ∣G/H∣=[G:H] or the index of H in G.