proof that |g| divides expG

The following is a proof that, for every group G that has an exponent and for every gG, |g| divides expG.


By the division algorithmPlanetmathPlanetmath, there exist q,r with 0r<|g| such that expG=q|g|+r. Since eG=gexpG=gq|g|+r=(g|g|)qgr=(eG)qgr=eGgr=gr, by definition of the order of an element, r cannot be positive. Thus, r=0. It follows that |g| divides expG. ∎

Title proof that |g| divides expG
Canonical name ProofThatgDividesoperatornameexpG
Date of creation 2013-03-22 13:30:35
Last modified on 2013-03-22 13:30:35
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 8
Author Wkbj79 (1863)
Entry type Proof
Classification msc 20D99