Center is normality-large implies every nontrivial normal subgroup contains a cyclic normal subgroup

Property-theoretic statement
The group property of being a group whose center is normality-large, is stronger than the group property of being a group in which every nontrivial normal subgroup contains a cyclic normal subgroup.

Verbal statement
If the center of a group has the property that its intersection with every nontrivial normal subgroup is nontrivial, then every nontrivial normal subgroup of the group contains a cyclic normal subgroup.