Characteristicity is strongly join-closed
This article gives the statement, and possibly proof, of a subgroup property (i.e., characteristic subgroup) satisfying a subgroup metaproperty (i.e., strongly join-closed subgroup property)
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Statement with symbols
Suppose is a group and is a (possibly empty) collection of characteristic subgroups of . Suppose the join of the s equals . By convention, the join of the empty collection is taken to be the trivial subgroup.
Then, is also a characteristic subgroup of .
Related facts about characteristicity
- Characteristicity is strongly intersection-closed
- Characteristicity does not satisfy intermediate subgroup condition
- Characteristicity is not upper join-closed
The statement has a generalization that states that any endo-invariance property is strongly join-closed. Here, endo-invarance means the proprty of being invariant under endomorphisms satisfying some given property. This fact, in turn, follows from the fact that homomorphisms commute with joins.
Other instances of the generalization are:
|Property||Endo-invariance property with respect to ...||Proof that it is strongly join-closed|
|Normal subgroup||inner automorphisms||Normality is strongly join-closed|
|Fully invariant subgroup||endomorphisms||Full invariance is strongly join-closed|
|Strictly characteristic subgroup||surjective endomorphisms||Strict characteristicity is strongly join-closed|
|Injective endomorphism-invariant subgroup||injective endomorphisms||Injective endomorphism-invariance is strongly join-closed|
- Closure-characteristicity is strongly join-closed
- Automorph-conjugacy is not finite-join-closed
- Procharacteristicity is not finite-join-closed