proof that expG divides |G|

The following is a proof that expG divides |G| for every finite groupMathworldPlanetmath G.


By the division algorithmPlanetmathPlanetmath, there exist q,r with 0r<expG such that |G|=q(expG)+r. Let gG. Then eG=g|G|=gq(expG)+r=gq(expG)gr=(gexpG)qgr=(eG)qgr=eGgr=gr. Thus, for every gG, gr=eG. By the definition of exponent, r cannot be positive. Thus, r=0. It follows that expG divides |G|. ∎

Title proof that expG 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