Nilpotent implies center is normality-large

Verbal statement
In a nilpotent group, the center is a normality-large subgroup; in other words, the intersection of the center with any nontrivial normal subgroup is a nontrivial normal subgroup.

Similar facts

 * Prime power order implies center is normality-large

Generalizations

 * Nilpotent implies intersection of normal subgroup with upper central series is strictly ascending till the subgroup is reached

Applications

 * Nilpotent and non-abelian implies center is not complemented
 * Minimal normal implies central in nilpotent
 * Socle equals Omega-1 of center in nilpotent p-group
 * Formula for number of minimal normal subgroups of group of prime power order
 * Congruence condition relating number of normal subgroups containing minimal normal subgroups and number of normal subgroups in the whole group
 * Thompson's critical subgroup theorem

Analogues in other algebraic structures

 * Nilpotent implies center is ideal-large

Proof
Given: A nilpotent group $$G$$ with center $$Z$$. A nontrivial normal subgroup $$N$$ of $$G$$.

To prove: $$N \cap Z$$ is nontrivial.

Proof: