If $G$ is a group and $y \in G$\ $\langle x \rangle$ , then can $\langle y \rangle$ be $G$?

Question

If $G$ is a group and $e \ne x \in G$ and $y \in G$\ $\langle x \rangle$ , then can $\langle y \rangle$ be $G$ ?

Answer 1

Have you tried $G = \mathbf{Z}$, $x = 2$ and $y = 1$ before asking ?

Answer 2

Let $G$ be cyclic (not necessarily finite), but with say $y$ as a generator. E.g. $G$ could be the infinite cyclic group, but with one generator.

Now $\langle y \rangle$ is by construction all of $G$, but not every element of $G$ is a generator. Pick $x$ in $G$ that is not a generator. For example, we can take $x=e$ the identity element.

Then $y \in G\setminus\langle x\rangle$, since if $y$ were generated by $x$, then $x$ would generate all of $G$.