Automorph-conjugate subgroup: Difference between revisions

Revision as of 06:19, 10 March 2007

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Definition

Symbol-free definition

A subgroup of a group is termed automorph-conjugate if it satsifies the following equivalent conditions:

Any subgroup to which it can go via an automorphism of the whole group, is also conjugate to the subgroup.
Whenever the bigger group is embedded as a normal subgroup of some ambient group, the product of the bigger group with the normalizer of the smaller group in the ambient group, is the whole group.

Definition with symbols

A subgroup $H$ of a group $G$ is termed automorph-conjugate if it satisfies the following equivalent conditions:

For any automorphism $σ$ of $G$ , $H$ and $σ (H)$ are conjugate subgroups in $G$ (that is, there exists $g \in G$ such that $σ (H) = g H g^{- 1}$ ).
Whenever $G ◃ M$ for some group $M$ , $G N_{M} (H) = M$ .

The latter formulation is important because it provides the necessary and sufficient conditions for Frattini's argument to go through.

Equivalence of definitions

Check out Frattini's argument.

In terms of the relation implication operator

The property of being automorph-conjugate can be viewed in terms of the relation implication operator with the relation on the left being that of a subgroup pair being automorphic and the relation on the right being that of a subgroup pair being conjugate in the whole group.

Relation with other properties

Stronger properties

Conjunction with other properties

Any normal automorph-conjugate subgroup is characteristic.

Metaproperties

Transitivity

The property of being automorph-conjugate does not appear to be transitive.

Trimness

The property of being isomorph-conjugate need not be identity-true. This is because a group may be isomorphic to a proper subgroup of itself, but they are clearly not conjugate in the group.

The property of being isomorph-conjugate is trivially true, that is, the trivial subgroup always satisfies the property in any group.

Intersection-closedness

Is the intersection of automorph-conjugate subgroups always automorph-conjugate?

Intermediate subgroup condition

The property of being automorph-conjugate does not satisfy the intermediate subgroup condition. The result of applying the intermediately operator to the property of being automorph-conjugate gives a property that, in particular, implies the property of being pronormal.

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 ===Symbol-free definition===
-A subgroup of a group is termed automorph-conjugate if any subgroup to which it can go via an automorphism of the whole group, is also conjugate to the subgroup.
+A subgroup of a group is termed automorph-conjugate if it satsifies the following equivalent conditions:
+* Any subgroup to which it can go via an automorphism of the whole group, is also conjugate to the subgroup.
+* Whenever the bigger group is embedded as a [[normal subgroup]] of some ambient group, the product of the bigger group with the normalizer of the smaller group in the ambient group, is the whole group.
 ===Definition with symbols===
-A subgroup <math>H</math> of a group <math>G</math> is termed automorph-conjugate if for any automorphism <math>\sigma</math> of <math>G</math>, <math>H</math> and <math>\sigma(H)</math> are conjugate subgroups in <math>G</math> (that is, there exists <math>g \in G</math> such that <math>\sigma(H) = gHg^{-1}</math>).
+A subgroup <math>H</math> of a group <math>G</math> is termed automorph-conjugate if it satisfies the following equivalent conditions:
+* For any automorphism <math>\sigma</math> of <math>G</math>, <math>H</math> and <math>\sigma(H)</math> are conjugate subgroups in <math>G</math> (that is, there exists <math>g \in G</math> such that <math>\sigma(H) = gHg^{-1}</math>).
+* Whenever <math>G \triangleleft M</math> for some group <math>M</math>, <math>GN_M(H) = M</math>.
+The latter formulation is important because it provides the necessary and sufficient conditions for [[Frattini's argument]] to go through.
+===Equivalence of definitions===
+Check out [[Frattini's argument]].
 ===In terms of the relation implication operator===