Abstract Algebra/Group Theory/Subgroup/Lagrange's Theorem

=Theorem= Let H be a subgroup of group G.

Let o(H), o(G), be orders of H, G respectively


 * o(H) divides o(G)

=Proof= As H is Subgroup of G,
 * 1. All Left Cosets of H partitions G.


 * 2. Each of such partitions is one of the Cosets of H.


 * 3. Any coset of H has the same order as H does.


 * 4. Thus, o(H) divides o(G)