Group Theory/Cosets and Lagrange's theorem

Right cosets are defined in an analogous fashion:

For both of these, we have the following proposition:

Analogously, we have the following proposition:

That is, the index is precisely the number of left cosets.

Hence, we may also use the notation $[G:H]$ for the number of right cosets.

Exercises

1. Prove that $g\in g^{\prime }H\iff gH=g^{\prime }H$, thus establishing another formula for the equivalence relation of being in the same coset.
2. Formulate Lagrange's theorem for right cosets, without using index notation.