existence of maximal subgroups

Because every finite group is a finite set, every chain of proper subgroupsof a finite group has a maximal element and thus every finite group hasa maximal subgroup. The same applies to maximal normal subgroups.

However, there are infinite groups, even abelian, with no maximal subgroups andno maximal normal subgroups. The Prüfer group

$${\mathbb{Z}}_{p^{\mathrm{\infty }}}=\underset{⟵}{lim}{\mathbb{Z}}_{p^i}$$

(for any prime $p$) is an example of an abelian group with no maximal subgroups.As the group is abelian all subgroups are normal so it also has no maximalnormal subgroups. Such groups fail to fit the hypothesis of the Jordan-Hölder decomposition theorem as they do not have the ascending chain condition and so we cannot assign a composition series to such groups.

This stands in contrast to the category of unital rings where if one assumes Zorn’s lemma (axiom of choice) then one may prove every unital ringhas a maximal ideal.