Center is strictly characteristic

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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 , with center . A surjective endomorphism .

To prove: .

Proof: Suppose . We need to show that for any , .

Since , there exists such that . Further, since is surjective, there exists such that . Since , we have:

.

Applying to both sides and using the property that is a homomorphism yields:

as required.