Generators of a cyclic group; how do you calculate them

The question reads: Find the number of generators of a cyclic group having the given order.

A) 8

B) 60

This is a practice question for the quiz. The answers are 4 and 16, but I’m confused how it’s calculated. Okay, so the cyclic group has a cardinality of 8. Do I need to figure out how many generators produce Z8? Z8 isn’t mentioned, but I don’t know what else to do.




Just a note. Z8\mathbb{Z}8 is the only cyclic group of order 8.
– baru
Oct 21 at 2:33


1 Answer


Recall Euler’s ϕ\phi function:
ϕ(n)=1\phi (n)=1 if n=1n=1 and ϕ(n)\phi (n) is the number of integers 1≤k≤n−11\leq k \leq n-1 such that (n,k)=1(n,k)=1 for n>1n>1.
Prove that if GG is a cyclic group with order nn, then the number of generators of GG is ϕ(n)\phi (n).