Center of normal implies normal

Statement
The center of any normal subgroup, is a normal subgroup of the whole group.

Applications
The result is sometimes used for induction in nilpotent groups, for instance, it is used for showing the following:


 * Equivalence of definitions of Fitting-free group: It shows that if there is a nontrivial nilpotent normal subgroup, there is a nontrivial Abelian normal subgroup
 * It is used in an argument to reduce the general case of the Schur-Zassenhaus theorem to the case of an Abelian Sylow subgroup.

Using characteristic subgroups
The proof pieces together two facts:


 * The center of any group is a characteristic subgroup (that's essentially because the center is a subgroup-defining function)
 * Any characteristic subgroup of a normal subgroup is normal