Center is strictly 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., strictly 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 always a strictly characteristic subgroup: any surjective endomorphism of the whole group sends the center to within itself.

Related facts

Stronger facts: stronger subgroup properties satisfied

Property	Meaning	Proof of satisfaction by the center	Proof that it is stronger than strict characteristicity
bound-word subgroup	described as a set of solutions to a system of equations, other variables quantified	center is bound-word	bound-word implies strictly characteristic

Stronger subgroup properties not satisfied

Property	Meaning	Proof of dissatisfaction by the center	Proof that it is stronger than strict characteristicity
normal-homomorph-containing subgroup	contains every homomorphic image that is normal in whole group	center not is normal-homomorph-containing	normal-homomorph-containing implies strictly characteristic
fully invariant subgroup	invariant under all endomorphisms	center not is fully invariant	fully invariant implies strictly characteristic

Weaker facts: weaker subgroup properties satisfied

(Logically, the center satisfies all weaker properties, but we single out some important ones here).

Property	Meaning	Proof of satisfaction by the center	Proof that it is weaker than strict characteristicity
characteristic subgroup	invariant under all automorphisms	center is characteristic	strictly characteristic implies characteristic
normal subgroup	invariant under all inner automorphisms	center is normal	(via characteristic)

Strict characteristicity for similar subgroup-defining functions

Subgroup-defining function (or collection of functions)	Meaning	Relation to center	Proof that it is strictly characteristic
member of upper central series	(transfinite) ascending series with each member's quotient by predecessor equal to center of group's quotient by predecessor	center is first member after trivial subgroup	upper central series members are strictly characteristic
Baer norm	intersection of normalizers of all subgroups	contains center, contained in second member of upper central series	Baer norm is strictly characteristic
Wielandt subgroup	intersection of normalizers of subnormal subgroup	contains center	Wielandt subgroup is strictly characteristic

Proof

Given: A group $G$ , with center $Z=Z(G)$ . A surjective endomorphism $\sigma :G\to G$ .

To prove: $\sigma (Z)\leq Z$ .

Proof: Suppose $g\in \sigma (Z)$ . We need to show that for any $h\in G$ , $gh=hg$ .

Since $g\in \sigma (Z)$ , there exists $a\in Z$ such that $g=\sigma (a)$ . Further, since $\sigma$ is surjective, there exists $b\in G$ such that $\sigma (b)=h$ . Since $a\in Z$ , we have: