Intermediately automorph-conjugate subgroup

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Definition

Symbol-free definition

A subgroup of a group is said to be intermediately automorph-conjugate if it is an automorph-conjugate subgroup in every intermediate subgroup (viz, every subgroup of the whole group containing it).

Definition with symbols

A subgroup $H$ of a group $G$ is said to be intermediately automorph-conjugate if for any subgroup $K$ of $G$ such that $H \le K$, $H$ is an automorph-conjugate subgroup of $K$. In other words, for any automorphism $\sigma$ of $K$, there exists $g \in K$ such that $\sigma(H) = gHg^{-1}$.

Formalisms

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In terms of the intermediately operator

This property is obtained by applying the intermediately operator to the property: automorph-conjugate subgroup
View other properties obtained by applying the intermediately operator

In terms of the intermediately operator

This property is obtained by applying the intermediately operator to the property: weakly procharacteristic subgroup
View other properties obtained by applying the intermediately operator

Relation with other properties

Stronger properties

• Isomorph-free subgroup
• Intermediately isomorph-conjugate subgroup
• Intermediately characteristic subgroup
• Sylow subgroup

Weaker properties

• Intermediately automorph-conjugate subgroup of normal subgroup
• Weakly pronormal subgroup
• Polynormal subgroup
• Intermediately normal-to-characteristic subgroup
• Intermediately subnormal-to-normal subgroup
• Automorph-conjugate subgroup