Automorph-conjugate subgroup

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Definition

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.

Definition with symbols

A subgroup $H$ of a group $G$ is termed automorph-conjugate if for any automorphism $\sigma$ of $G$, $H$ and $\sigma (H)$ are conjugate subgroups in $G$ (that is, there exists $g\in G$ such that $\sigma (H)=gH{g}^{-1}$).

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

• Isomorph-conjugate subgroup
• Characteristic subgroup
• Sylow subgroup

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.