Center is normal

Statement
The center of any group is a normal subgroup.

More about the center

 * The quotient group of the group by its center is isomorphic to the inner automorphism group. This is because the center is the kernel of the group action on itself by conjugation, whose image is the inner automorphism group.

Stronger subgroup properties satisfied

 * Hereditarily normal subgroup: Every subgroup of the center is a normal subgroup.
 * Central subgroup, central factor, transitively normal subgroup
 * Characteristic subgroup:
 * Strictly characteristic subgroup:
 * Bound-word subgroup:

Stronger subgroup properties not satisfied

 * Fully invariant subgroup:
 * Injective endomorphism-invariant subgroup:
 * Weakly normal-homomorph-containing subgroup: