TITLE: Why are all cyclic groups countable?
QUESTION [6 upvotes]: I'm new to group theory. Why are all cyclic groups countable? And does countable mean finite or denumerable?

REPLY [2 votes]: I would say that this is because the word generate here means the repeated or iterated group operation, i.e., a group $(G,*)$ is generated by a set $S$ ({$S=g$} here) if every element $g$ in $G$ can be obtained by the repeated operation of g with itself. This only makes sense for $1, 2,3$, etc. iterations or repeated operations; operating g with itself a non-integer number of times does not have a clear meaning unless you deal with, e.g., topological groups. In this context you have, e.g., 1-parameter groups you may in some sense say are cyclic: http://en.wikipedia.org/wiki/One-parameter_group