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 $$g \in g'H \Leftrightarrow gH = g'H$$, 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.