Lagrange's Theorem 라그랑지 정리

" If G is a finite group and H ≤ G , then |H| divides |G| "

[Proof]

[Lemma] For any G (finite or infinite)  , Every coset of H (≤ G) has the same cardinality.
because ∃a one-to-one map Φ from H onto gH ∀g∈G.

For example, Φ(h) = gh for each h∈H.
onto : clear by definition of Φ.
one-to-one : by cancellation law.
Thus, ∀g∈G , |gH| = |Hg| = |H|

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Since cosets are the cells in the partition by an equivalence relation in G, cosets are disjoint subsets of G.
Therefore,  ∀g∈G,  ∑ |gH| = ∑ |Hg| = |G|

Let G be a finite group and r be the number of cells in the partition of G into left(or right) cosets of H.
then ∑ |gH| = ∑ |Hg| = r|H| = |G| and it follows that " |H| divides |G| ."