generator


If G is a cyclic groupMathworldPlanetmath and gG, then g is a generatorPlanetmathPlanetmathPlanetmath of G if g=G.

All infinite cyclic groups have exactly 2 generators. To see this, let G be an infinite cyclic group and g be a generator of G. Let z such that gz is a generator of G. Then gz=G. Then gG=gz. Thus, there exists n with g=(gz)n=gnz. Therefore, gnz-1=eG. Since G is infiniteMathworldPlanetmath and |g|=|g|=|G| must be infinityMathworldPlanetmath, nz-1=0. Since nz=1 and n and z are integers, either n=z=1 or n=z=-1. It follows that the only generators of G are g and g-1.

A finite cyclic group of order n has exactly φ(n) generators, where φ is the Euler totient function. To see this, let G be a finite cyclic group of order n and g be a generator of G. Then |g|=|g|=|G|=n. Let z such that gz is a generator of G. By the division algorithmPlanetmathPlanetmath, there exist q,r with 0r<n such that z=qn+r. Thus, gz=gqn+r=gqngr=(gn)qgr=(eG)qgr=eGgr=gr. Since gr is a generator of G, it must be the case that gr=G. Thus, n=|G|=|gr|=|gr|=|g|gcd(r,|g|)=ngcd(r,n). Therefore, gcd(r,n)=1, and the result follows.

Title generator
Canonical name Generator
Date of creation 2013-03-22 13:30:39
Last modified on 2013-03-22 13:30:39
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 10
Author Wkbj79 (1863)
Entry type Definition
Classification msc 20A05
Related topic GeneratingSetOfAGroup
Related topic ProperGeneratorTheorem