Center of normal implies normal

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This fact is an application of the following pivotal fact/result/idea: characteristic of normal implies normal
View other applications of characteristic of normal implies normal OR Read a survey article on applying characteristic 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:

Proof

Using characteristic subgroups

The proof pieces together two facts: