Fully invariant implies characteristic: Difference between revisions

Latest revision as of 20:01, 9 October 2009

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., fully invariant subgroup) must also satisfy the second subgroup property (i.e., characteristic subgroup)
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Get more facts about fully invariant subgroup|Get more facts about characteristic subgroup

Statement

Any fully invariant subgroup (also called fully characteristic subgroup) of a group is a characteristic subgroup.

Definitions used

Fully invariant subgroup

Further information: fully invariant subgroup

A subgroup $H$ of a group $G$ is termed fully invariant in $G$ if, for every endomorphism $φ$ of $G$ , $φ (H)$ is contained in $H$ .

Characteristic subgroup

Further information: characteristic subgroup

A subgroup $H$ of a group $G$ is termed characteristic in $G$ if, for every automorphism $φ$ of $G$ , $φ (H)$ is contained in $H$ .

Related facts

Intermediate properties

For intermediate notions between characteristic subgroup and fully invariant subgroup, click here.

Invariance under restricted classes of endomorphisms

Strictly characteristic subgroup is a subgroup that is invariant under all surjective endomorphisms. Every fully invariant subgroup is strictly characteristic, and every strictly characteristic subgroup is characteristic.
Injective endomorphism-invariant subgroup is a subgroup that is invariant under all injective endomorphisms. Every fully invariant subgroup is strictly characteristic, and every strictly characteristic subgroup is characteristic.

Proof

Proof idea

The idea behind the proof is that since every automorphism is an endomorphism, invariance under all endomorphisms implies invariance under automorphisms.

Using function restriction expressions

This subgroup property implication can be proved by using function restriction expressions for the subgroup properties
View other implications proved this way |read a survey article on the topic

The property of being fully invariant is the invariance property with respect to endomorphisms. It has the function restriction expression:

Endomorphism $\to$ Function

The property of being characteristic is the invariance property with respect to automorphisms. It has the function restriction expression:

Automorphism $\to$ Function

Since the left side of the function restriction expression for characteristicity is stronger while the right sides of both function restriction expressions are equal, the property of being characteristic is weaker than the property of being fully invariant.

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 A subgroup <math>H</math> of a group <math>G</math> is termed characteristic in <math>G</math> if, for every [[automorphism]] <math>\varphi</math> of <math>G</math>, <math>\varphi(H)</math> is contained in <math>H</math>.
+==Related facts==
+* [[Characteristic not implies fully invariant]]
+* [[Characteristic not implies fully invariant in finite abelian group]]
+* [[Characteristic equals fully invariant in odd-order abelian group]]
+==Intermediate properties==
+{{intermediate notions|characteristic subgroup|fully invariant subgroup}}
+===Invariance under restricted classes of endomorphisms===
+* [[Strictly characteristic subgroup]] is a subgroup that is invariant under all [[surjective endomorphism]]s. Every fully invariant subgroup is strictly characteristic, and every strictly characteristic subgroup is characteristic.
+* [[Injective endomorphism-invariant subgroup]] is a subgroup that is invariant under all [[injective endomorphism]]s. Every fully invariant subgroup is strictly characteristic, and every strictly characteristic subgroup is characteristic.
 ==Proof==