Equivalence of definitions of finitely generated group

Statement
The following are equivalent for a group $$G$$:


 * 1) It has a finite generating set.
 * 2) Every generating set of the group has a subset that is finite and is also a generating set.
 * 3) The group has at least one minimal generating set and every minimal generating set of the group is finite.
 * 4) The minimum size of generating set of the group is finite.
 * 5) The group is a join of finitely many cyclic subgroups.

(1) implies (2)
Given: A group $$G$$ with a finite generating set $$A$$ and a generating set $$B$$ (not necessarily finite).

To prove: There exists a finite subset of $$B$$ that is also a generating set for $$G$$.

Proof:

(2) implies (3)
Given: A group $$G$$ with the property that every generating set for $$G$$ contains a finite subset that is also a generating set.

To prove: $$G$$ has at least one minimal generating set, and every minimal generating set of the group is finite.

Proof:

Proof of the first part: Start with $$G$$ as a generating set for itself. By the given, there exists a subset $$C \subseteq G$$ that is finite and is a generating set for $$G$$. Let $$\mathcal{F}$$ be the collection of subsets of $$C$$ that generate $$G$$. $$\mathcal{F}$$ is nonempty since $$C \in \mathcal{F}$$. Since it is finite, it must have a minimal element. Thus, there exists a minimal generating set for $$G$$.

Proof of the second part: Suppose $$D$$ is a generating set for $$G$$. If $$D$$ is infinite, then it cannot be minimal, since by assumption, it contains a finite (and hence strictly smaller) generating set for $$G$$. Thus, $$D$$ must be finite.

(3) implies (4)
This is direct.

(4) implies (1)
This is direct.

(1) implies (5)
For any finite generating set, the group is the union of the cyclic subgroups generated by the individual generators.

(5) implies (1)
If the group is a union of finitely many cyclic subgroups, pick one (cyclic) generator for each cyclic subgroup. The set comprising these generators is a finite generating set for the group.