# Isomorph-normal subgroup

From Groupprops

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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]

## Definition

A subgroup of a group is termed an **isomorph-normal subgroup** if every subgroup of the group isomorphic to it is a normal subgroup of the whole group.

## Relation with other properties

### Stronger properties

- Isomorph-free subgroup
- Isomorph-characteristic subgroup
- Isomorph-normal characteristic subgroup
- Maximal subgroup of finite nilpotent group
- Order-normal subgroup

### Weaker properties

## Metaproperties

### Join-closedness

YES:This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property.In fact, since the property is also true for the trivial subgroup in any group, it is a strongly join-closed subgroup property.ABOUT THIS PROPERTY: View variations of this property that are join-closed | View variations of this property that are not join-closedABOUT JOIN-CLOSEDNESS: View all join-closed subgroup properties (or, strongly join-closed properties) | View all subgroup properties that are not join-closed | Read a survey article on proving join-closedness | Read a survey article on disproving join-closedness

An arbitrary join of isomorph-normal subgroups is isomorph-normal. `For full proof, refer: Isomorph-normality is strongly join-closed`

### Intersection-closedness

This subgroup property is not intersection-closed, viz., it is not true that an intersection of subgroups with this property must have this property.

Read an article on methods to prove that a subgroup property is not intersection-closed

An intersection of isomorph-normal subgroups need not be isomorph-normal.