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.

Related facts

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