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.
Best Answer
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.