Center is characteristic

This article gives the statement, and possibly proof, of the fact that for any group, the subgroup obtained by applying a given subgroup-defining function (i.e., center) always satisfies a particular subgroup property (i.e., characteristic subgroup)}
View subgroup property satisfactions for subgroup-defining functions $|$ View subgroup property dissatisfactions for subgroup-defining functions

Statement

The center of a group is a characteristic subgroup of it.

Related facts

Stronger facts: stronger subgroup properties satisfied

The center satisfies a number of stronger properties:

Property	Meaning	Proof of satisfaction by center	Proof that it's stronger than characteristicity
strictly characteristic subgroup	invariant under all surjective endomorphisms	Center is strictly characteristic	strictly characteristic implies characteristic
quasiautomorphism-invariant subgroup	invariant under all quasiautomorphisms	Center is quasiautomorphism-invariant	quasiautomorphism-invariant implies characteristic
purely definable subgroup	can be defined in first-order language using the pure theory of the group	Center is purely definable	purely definable implies characteristic
elementarily characteristic subgroup	no other elementarily equivalently embedded subgroups	Center is elementarily characteristic	elementarily characteristic implies characteristic
bound-word subgroup	described as set of solutions of a system of equations, other variables quantified	Center is bound-word	(via strictly characteristic)

Stronger subgroup properties not satisfied

Property	Meaning	Proof of dissatisfaction	Proof that it is stronger than characteristicity
fully invariant subgroup	invariant under all endomorphisms	center not is fully invariant	fully invariant implies characteristic
normal-homomorph-containing subgroup	contains any homomorphic image that is normal in whole group	center not is normal-homomorph-containing	(via strictly characteristic)
weakly normal-homomorph-containing subgroup	contains any homomorphic image in a map that sends normal subgroups to normal subgroups	center not is weakly normal-homomorph-containing	(via strictly characteristic)
1-automorphism-invariant subgroup	invariant under all 1-automorphisms	center not is 1-automorphism-invariant	1-automorphism-invariant implies characteristic

Weaker facts: weaker subgroup properties satisfied

• Center is normal

Analogues in other algebraic structures

• Center is characteristic for Lie rings
• Center is derivation-invariant (another analogue for Lie rings)