generator (original) (raw)
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∈ℤ such that gz is a generator of G. Then ⟨gz⟩=G. Then g∈G=⟨gz⟩. Thus, there exists n∈ℤ with g=(gz)n=gnz. Therefore, gnz-1=eG. Since G is infinite
and |g|=|⟨g⟩|=|G| must be infinity
, 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∈ℤ such that gz is a generator of G. By the division algorithm, there exist q,r∈ℤ with 0≤r<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.