# Intermediately isomorph-conjugate subgroup

From Groupprops

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## Contents

## 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 of a group is termed **intermediately isomorph-conjugate** if given any other subgroup of which is isomorphic to , there is an element such that .

## Formalisms

BEWARE!This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

### 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

### 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 conditionABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

If is intermediately isomorph-conjugate in , then 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 are intermediately isomorph-conjugate subgroups, and , then the join of subgroups is also an intermediately isomorph-conjugate subgroup. `Further information: Intermediate isomorph-conjugacy is normalizing join-closed`