Maximal normal subgroup ($N⊲G,N\ne G,N\subseteq M⊲G\Rightarrow M=N\text{ or }M=G$)

A proper normal subgroup maximal among proper normal subgroups. Its quotient is a nontrivial simple group. Maximality among normal subgroups does not imply maximality among all subgroups.

// Target: algebra.bigb

Above any proper normal subgroup of a finitely generated group, a chain of proper normal subgroups has proper union. Otherwise that union would contain the finite generating set, all of which would already lie in one member of the chain. The Zorn lemma therefore provides a maximal proper normal subgroup containing the given one.

// Target: algebra.bigb