Prove: Every cyclic group with order > 2 has at least 2 distinct generators
Here's what I've got so far:
Either order of our group, $G$, is finite or infinite:
Suppose infinite: then our group is isomorphic to $Z$ under addition. This group has two distinct generators, therefore so does G.
Suppose finite: Then $G$ is isomorphic to $Z$n under addition modulo $n$. We know order of $G$ is >= 3. If order of G is odd, both 1 and 2 are generators of $Z$n under addition modulo $n$, so $G$ has at least two generators.
If order of $G$ is even, then both 1 and some odd number between 1 and $n$ are generators. How do we determine this odd number?
Does this look like I'm on the right track?
Thanks for the help, Mariogs