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[parent] proof of Lagrange's theorem (Proof)

We know that the cosets $Hg$ form a partition of $G$ (see the coset entry for proof of this.) Since $G$ is finite, we know it can be completely decomposed into a finite number of cosets. Call this number $n$ We denote the $i$ coset by $Ha_i$ and write $G$ as

$$ G = Ha_1 \cup Ha_2 \cup \cdots \cup Ha_n $$

since each coset has $|H|$ elements, we have

$$ |G| = |H|\cdot n $$

and so $|H|$ divides $|G|$ which proves Lagrange's theorem. $\square$



"proof of Lagrange's theorem" is owned by akrowne.
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Cross-references: Lagrange's theorem, divides, number, finite, proof, partition, cosets

This is version 2 of proof of Lagrange's theorem, born on 2002-02-02, modified 2002-02-03.
Object id is 1663, canonical name is ProofOfLagrangesTheorem.
Accessed 10963 times total.

Classification:
AMS MSC20D99 (Group theory and generalizations :: Abstract finite groups :: Miscellaneous)

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