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.

Answer 2

We can - who said we couldn't? You can even have irredundant generating sets with more than one element. For example $C_6 = \langle g \rangle = \langle g^2,g^3 \rangle$, where $g$ has order $6$.

Answer 3

A set with more than one element can be a generator of a cyclic group. Consider for example the (additive) group $\mathbb{Z}/6\mathbb{Z}$. It is cyclic, because it is generated by the set {1}. But it is also generated by any of the sets {1, 3, 5} or {2, 3}. The important part is that it is possible th generate the group by a singleton.

As a counterexample consider the group $(\mathbb{Z}/2\mathbb{Z}) \times (\mathbb{Z}/2\mathbb{Z})$ which is not cyclic.