Let H be a group acting on a set A. Prove that the relation ~ on A defined by a~b iff for some h in H is an equivalence relation.

I have already shown this.

Let H be a subgroup of the finite group G and let H act on G by left multiplication. Let x exist in G and let O be the orbit of x under the action of H. Prove that the map defined by is a bijection.

I have already shown this too.

From these two statements, deduce Lagrange's Theorem: if G is a finite group and , then .

I understand Lagrange's Theorem. I need an explanation on how I can deduce his theorem from the above.