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

# Theorem

editLet H be a subgroup of group G.

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

- o(H) divides o(G)

# Proof

editAs 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)