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

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.

Corollaries

 * Minimal normal implies contained in Omega-1 of center for nilpotent p-group
 * Socle equals Omega-1 of center for nilpotent p-group
 * Minimal characteristic implies contained in Omega-1 of center for nilpotent p-group

Other related facts

 * Minimal normal implies central in nilpotent
 * Minimal characteristic implies central in nilpotent

Facts used

 * 1) Nilpotent implies center is normality-large
 * 2) Omega-1 is large (and hence, is normality-large)
 * 3)  Normality-largeness is transitive