proof that $\exp G$ divides $|G|$

The following is a proof that $\exp G$ divides $|G|$ for every finite group $G$.

Proof.

By the division algorithm, there exist $q,r\in \mathbb{Z}$ with such that $|G|=q(\exp G)+r$. Let $g\in G$. Then $e_G=g^{|G|}=g^{q(\exp G)+r}=g^{q(\exp G)}{g}^r={(g^{\exp G})}^q{g}^r={(e_G)}^q{g}^r=e_G{g}^r=g^r$. Thus, for every $g\in G$, $g^r=e_G$. By the definition of exponent, $r$ cannot be positive. Thus, $r=0$. It follows that $\exp G$ divides $|G|$. ∎

Title	proof that $\exp G$ divides $|G|$
Canonical name	ProofThatoperatornameexpGDividesG
Date of creation	2013-03-22 13:30:32
Last modified on	2013-03-22 13:30:32
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	10
Author	Wkbj79 (1863)
Entry type	Proof
Classification	msc 20D99