Minimal normal implies contained in Omega-1 of center for nilpotent p-group

Statement
In a fact about::nilpotent p-group $$G$$, any fact about::minimal normal subgroup is contained in the subgroup $$\Omega_1(Z(G))$$ (the first omega subgroup of the center of $$G$$). Thus, the fact about::socle of $$G$$ is contained in $$\Omega_1(Z(G))$$. (in fact, something stronger is true: for a nilpotent p-group, the socle equals Omega-1 of the center).

Related facts

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

Facts used

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