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
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|
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: