That's right--the inverse of a generator is a generator, and a generator is never its own inverse unless it's the only nonidentity element.
My problem is to prove that cyclic groups of order 3 or greater must have at least 2 generators.
I think I found an answer but I'm worried that it's wrong.
Here's my proof.
In a group of order generated by there exists a "last element" . This element can be shown to generate every inverse element and therefore every element in the group.
For example
and so on until
Thus it is shown that in every cyclic group with the element generates the group as well as .
Is this valid?
Right, except I'm not sure if you're allowed to assume the cyclic group is finite. (I'm not sure if saying is supposed to mean that is finite or not.) In this situation, though, just replace with (which, in the case of a finite cyclic group, are the same thing).