Characteristicity is strongly join-closed: Difference between revisions

Revision as of 01:16, 16 January 2010

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)
View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about characteristic subgroup |Get facts that use property satisfaction of characteristic subgroup | Get facts that use property satisfaction of characteristic subgroup|Get more facts about strongly join-closed subgroup property

Statement

Statement with symbols

Suppose $G$ is a group and $H_{i},i\in I$ is a (possibly empty) collection of characteristic subgroups of $G$ . Suppose the join of the $H_{i}$ s equals $H$ . By convention, the join of the empty collection is taken to be the trivial subgroup.

Then, $H$ is also a characteristic subgroup of $G$ .

Related facts

Related facts about characteristicity

Generalizations

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

Join-closedness for related properties

Proof

PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

@@ Line 2: / Line 2: @@
 property = characteristic subgroup|
 metaproperty = strongly join-closed subgroup property}}
+==Statement==
+===Statement with symbols===
+Suppose <math>G</math> is a [[group]] and <math>H_i, i \in I</math> is a (possibly empty) collection of [[characteristic subgroup]]s of <math>G</math>. Suppose the [[join of subgroups|join]] of the <math>H_i</math>s equals <math>H</math>. By convention, the join of the empty collection is taken to be the trivial subgroup.
+Then, <math>H</math> is also a [[characteristic subgroup]] of <math>G</math>.
+==Related facts==
+===Related facts about characteristicity===
+* [[Characteristicity is strongly intersection-closed]]
+* [[Characteristicity does not satisfy intermediate subgroup condition]]
+* [[Characteristicity is not upper join-closed]]
+===Generalizations===
+The statement has a generalization that states that any [[endo-invariance implies strongly join-closed|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:
+{| class="sortable" border="1"
+! Property !! Endo-invariance property with respect to ... !! Proof that it is strongly join-closed
+|-
+| [[Normal subgroup]] || [[inner automorphism]]s || [[Normality is strongly join-closed]]
+|-
+| [[Fully invariant subgroup]] || [[endomorphism]]s || [[Full invariance is strongly join-closed]]
+|-
+| [[Strictly characteristic subgroup]] || [[surjective endomorphism]]s || [[Strict characteristicity is strongly join-closed]]
+|-
+| [[Injective endomorphism-invariant subgroup]] || [[injective endomorphism]]s || [[Injective endomorphism-invariance is strongly join-closed]]
+|}
+===Join-closedness for related properties===
+* [[Closure-characteristicity is strongly join-closed]]
+* [[Automorph-conjugacy is not finite-join-closed]]
+* [[Procharacteristicity is not finite-join-closed]]
+==Proof==
+{{fillin}}