[Math] Why is the cyclic group just generated by a single element

Question

I know the definition of a cyclic group after I learned a group generated by a set. A cyclic group is a group generated by a single element. But why we can't generate a cyclic group by a set that has more than one element?

Answer 1

When you read a compact phrase like

a group generated by a single element

You should interpret that to mean:

a group such that there exists a single-element set that generates the group

But not:

a group whose generating set has a single element

The latter interpretation is syntactically sound, and it sounds plausible if you haven't studied groups before. But it implicitly assumes that a group has a distinguished generating set, so that you can speak of "the" generating set of a group. Generally speaking, this is not the case. So the former interpretation is the only one that makes sense.

Indeed, the generating sets, plural, of a group can be very different from each other. See the other answers for examples.