Extensible automorphism-invariant equals normal: Difference between revisions

Latest revision as of 23:41, 10 October 2008

This article gives a proof/explanation of the equivalence of multiple definitions for the term normal subgroup
View a complete list of pages giving proofs of equivalence of definitions

Statement

The following are equivalent for a subgroup of a group:

The subgroup is a normal subgroup: it is invariant under all inner automorphisms of the whole group.
The subgroup is an extensible automorphism-invariant subgroup: it is invariant under all extensible automorphisms of the whole group.

Facts used

Proof

Conceptual proof using function restriction expressions

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

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-{{subgroup property}}
+{{definition equivalence|normal subgroup}}
-{{finitarily equivalent to|normality}}
+==Statement==
-{{wikilocal}}
+The following are equivalent for a subgroup of a group:
-==Definition==
+* The subgroup is a [[normal subgroup]]: it is invariant under all [[inner automorphism]]s of the whole group.
+* The subgroup is an '''extensible automorphism-invariant subgroup''': it is invariant under all [[extensible automorphism]]s of the whole group.
-===Symbol-free definition===
+==Facts used==
-A [[subgroup]] of a [[group]] is termed '''extensible automorphism-invariant''' if any [[extensible automorphism]] on the whole group takes the subgroup to within itself.
+# [[uses::Inner implies extensible]]
+# [[uses::Extensible implies normal]]
-Here an extensible automorphism means an automorphism of the group that can be extended to an automorphism for any group containing it.
+==Proof==
-===Definition with symbols===
+===Conceptual proof using function restriction expressions===
-A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''extensible automorphism-invariant''' if for any extensible automorphism <math>\sigma</math> of <math>G</math>, <math>\sigma(H) \subseteq H</math>.
+{{fillin}}
-==Formalisms==
-The subgroup property of being extensible automorphism-invariant can be expressed using the [[function restriction formalism]] in the following ways:
-* As the {{invariance property}} with respect to the function property of being an extensible automorphism, viz:
-[[Extensible automorphism]] <math>\to</math> Function
-* As the function restriction formal expression with the left side being extensible automorphisms and the right side being automorphisms, viz:
-[[Extensible automorphism]] <math>\to</math> [[Automorphism]]
-Further, the latter expression is [[right tight restriction foraml expression|right tight]], viz we cannot replace automorphism by anything stronger. This follows by a similar reasoning to that for right tightness for the corresponding expression of normality.
-==Relation with other properties==
-===Stronger properties===
-* [[Strongly potentially characteristic subgroup]]
-* [[Potentially characteristic subgroup]]
-* [[Potentially relatively characteristic subgroup]]
-* [[Pushforwardable automorphism-invariant subgroup]]
-===Weaker properties===
-* [[Normal subgroup]]
-===Conjecture of equalling normality===
-{{conjecturedtoequal|normality}}
-The conjecture has already been settled for groups that are finite or that haev some finiteness conditions. Note that this is equivalent to the conjecture that every extensible automorphism is [[normal automorphism|normal]].
-==Metaproperties==