Center of normal implies normal

This fact is an application of the following pivotal fact/result/idea: characteristic of normal implies normal
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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.

Proof

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