On finite exponent abelian $p$-groups

Question

Let $G$ be an abelian $p$-group non-isomorphic to any group of the form $H\times K$ where $H$ and $K$ are nontrivial groups. And let $\{|a|\mid a\in G\}$ have an upper bound in $\Bbb N$ . Is $G$ finitely generated?

Answer 1

This is Theorem 17.2 in Fuchs' "Infinite Abelian Groups" (vol.1), the result is originally due to Prüfer and Baer:

Theorem. A bounded Abelian group is a direct sum of cyclic groups.

As a corollary you obtain that any indecomposable bounded Abelian group is cyclic.

Answer 2

The answer would seem to be no. A finitely generated Abelian group is a direct product of finitely many cyclic groups (some of which may be infinite). In your situation, the cyclic factors would be of bounded order, so if your group were finitely generated, it would be finite (and even cyclic by the indecomposability condition you impose).

Later edit: In view of comments, yes, I overlooked one case. The group in question is either finite cyclic of prime power order (such a group is not a direct product of smaller proper cyclic subgroups), or it is infinitely generated.