## Every nontrivial finitely generated group has maximal subgroups

This is an exercise using Zorn’s Lemma to prove that every nontrivial finitely generated group has maximal subgroups. Let be a finitely generated group, say , and let be the set of all proper subgroups of . is partially ordered by . Let be an arbitrary chain in .

- Prove that is a subgroup of .
- Prove that is a
*proper* subgroup of .
- Apply Zorn’s Lemma to infer the existence of a maximal subgroup.

- Let . Now we have and for some , in . Since is a chain, either or . Without loss of generality, suppose . Then , so that . By the Subgroup Criterion, is a subgroup of .
- Suppose . Then for each generator , we have for some integer . Recall that every finite set of integers has a greatest element; say the greatest element of is . Then for all , we have . But then , so that is not proper. This is a contradiction, so is a proper subgroup of .
- By the previous two parts, every chain in has an upper bound in . By Zorn’s Lemma, has a maximal element which is by definition a maximal subgroup of .

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