Center is 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., 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

Analogues in other algebraic structures