Omega-1 of center is normality-large in nilpotent p-group

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Statement

Let G be a nilpotent p-group, i.e., a nilpotent group where the order of every element is a power of the prime p. Then, the subgroup \Omega_1(Z(G)) is a normality-large subgroup of G: its intersection with every nontrivial normal subgroup is nontrivial.

Here, \Omega_1 denotes the omega subgroup: the subgroup generated by all the elements of order p, and Z(G) denotes the center of G.

Note that if G is a finite p-group, i.e., a group of prime power order, then it is nilpotent.

Facts used