TITLE: How to find a generator of a cyclic group?
QUESTION [43 upvotes]: A cyclic group is a group that is generated by a single element.  That means that there exists an element $g$, say, such that every other element of the group can be written as a power of $g$.  This element $g$ is the generator of the group.
Is that a correct explanation for what a cyclic group and a generator are? How can we find the generator of a cyclic group and how can we say how many generators should there be?

REPLY [6 votes]: 1.) For large group orders it is no suitable to explicitly evaluate all powers of an optional generator to prove the element's order.
2.) The fact that a multiplicative cyclic finite group is isomorphic to some additive finite subgroup in ℤ is not helpful, as the isomorphism is defined exactly by a generator.
Criterion: An element $g$ of multiplicative group of order $(p-1)$ in $ℤ/pℤ$  with prime $p$ is a generator, iff for each prime factor $q$ in the factorization of $p-1$
     g^((p-1)/q) <> 1

holds.
This excludes $g$ from being generator of a real subgroup and
reduces the problem to factorization of $p-1$.