proof of Lagrange's theorem
ProofOfLagrangesTheorem
2002-02-02 18:32:04
2002-02-03 00:44:24
Proof
Lagrange's theorem
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
%\usepackage{psfrag}
%\usepackage{graphicx}
%\usepackage{xypic}
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. $\square$