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

Statement

In a Nilpotent p-group (?) $G$, any Minimal normal subgroup (?) is contained in the subgroup $\Omega _{1}(Z(G))$ (the first omega subgroup of the center of $G$). Thus, the 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