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