proof of Lagrange’s theorem

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 $i$th 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. $\square$

Title proof of Lagrange’s theorem ProofOfLagrangesTheorem 2013-03-22 12:15:47 2013-03-22 12:15:47 akrowne (2) akrowne (2) 6 akrowne (2) Proof msc 20D99