Intermediately isomorph-conjugate subgroup

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Definition

Symbol-free definition

A subgroup of a group is termed intermediately isomorph-conjugate if it is isomorph-conjugate in every intermediate subgroup.

Definition with symbols

A subgroup $H$ of a group $G$ is termed intermediately isomorph-conjugate if given any other subgroup $L$ of $G$ which is isomorphic to $H$, there is an element $x\in <H,L>$ such that $xHx^{-1}=L$.

Formalisms

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In terms of the intermediately operator

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

Relation with other properties

Stronger properties

• Sylow subgroup in a finite group
• Hall subgroup in a finite solvable group
• Isomorph-free subgroup

Weaker properties

• Intermediately isomorph-conjugate subgroup of normal subgroup
• Pronormal subgroup
• Weakly pronormal subgroup
• Paranormal subgroup
• Polynormal subgroup
• Intermediately automorph-conjugate subgroup
• Intermediately normal-to-characteristic subgroup
• Intermediately subnormal-to-normal subgroup
• Isomorph-conjugate subgroup

Metaproperties

Intermediate subgroup condition

YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

If $H$ is intermediately isomorph-conjugate in $G$, then $H$ is also intermediately isomorph-conjugate in any intermediate subgroup. This follows from the definition.

Normalizing joins

This subgroup property is normalizing join-closed: the join of two subgroups with the property, one of which normalizes the other, also has the property.
View other normalizing join-closed subgroup properties

If $H,K\leq G$ are intermediately isomorph-conjugate subgroups, and $K\leq N_{G}(H)$, then the join of subgroups $HK$ is also an intermediately isomorph-conjugate subgroup. Further information: Intermediate isomorph-conjugacy is normalizing join-closed