Nilpotent implies every nontrivial normal subgroup contains a cyclic normal subgroup

Statement
In a nilpotent group, every nontrivial normal subgroup contains a nontrivial cyclic normal subgroup.

Facts used

 * 1) uses::Nilpotent implies center is normality-large
 * 2) uses::Center is normality-large implies every nontrivial normal subgroup contains a cyclic normal subgroup