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)