Generating set of an infinite group

Question

Must a generating set for a group of infinite order be finite?

E.g $\{1\}$ is a finite generating set for $(\mathbb{Z},+)$.

Answer 1

$\mathbb{Z}$ is the free group on one generator. A free group is completely determined by its generating set, which can be any set. So we can consider a free group on an infinite number of generators as a counterexample.

For a more concrete example consider $(\mathbb{R},+)$. If this were finitely generated then we would have $|\mathbb{R}|\leq|x_1\mathbb{Z}|\cdot|x_2\mathbb{Z}|\cdots|x_n\mathbb{Z}|$. This is a contradiction.