Does any infinitely generated group have no maximal subgroups?

Question

I've known that every finite group has maximal subgroups. And, use Zorn's Lemma, every finitely generated group has maximal subgroups.

Also, there are examples that have no maximal subgroups, like $(\mathbb{Q},+)$ or the Prüfer group $\mathbb{Z}(p^\infty)$. They are both infinitely generated.

So I have this question that is every infinite generated group has no maximal subgroups? Or is there any counterexample?

Appreciate any suggestions in advance.

Answer 1

The group of nonzero real numbers under multiplication is not finitely generated and has a maximal proper subgroup consisting of the positive numbers.

Answer 2

Given any group $G$ and a finite group $H$ we have that $G\times H$ has a maximal subgroup. More precisely $G\times M$ is such subgroup for a maximal subgroup $M$ in $H$.

More generally the same will apply to any group $G$ that has a finite quotient $G/H$. Because there is one-to-one correspondence (which preserves inclusions) between subgroups of $G/H$ and subgroups of $G$ containing $H$. In particular every free group has a maximal subgroups, regardless of rank or cardinality.