Nilpotent implies every nontrivial normal subgroup contains a cyclic normal subgroup

This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., nilpotent group) must also satisfy the second group property (i.e., group in which every nontrivial normal subgroup contains a cyclic normal subgroup)
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Statement

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

Facts used

1. Nilpotent implies center is normality-large
2. Center is normality-large implies every nontrivial normal subgroup contains a cyclic normal subgroup