# Intermediately normal-to-characteristic 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

### Symbol-free definition

A subgroup of a group is termed **intermediately normal-to-characteristic** if it satisfies the following equivalent conditions:

- Whenever it is normal in any intermediate subgroup, it is also characteristic in the intermediate subgroup.
- It is an intermediately characteristic subgroup in its normalizer.

### Definition with symbols

A subgroup of a group is termed **intermediately normal-to-characteristic** in if it satisfies the following equivalent conditions:

- For any subgroup of containing such that is normal in , is characteristic in .
- is characteristic in any subgroup of contained in its normalizer .

## Formalisms

### In terms of the in-normalizer operator

This property is obtained by applying the in-normalizer operator to the property: intermediately characteristic subgroup

View other properties obtained by applying the in-normalizer operator

A subgroup is intermediately normal-to-characteristic in if and only if is an intermediately characteristic subgroup in .

## Relation with other properties

### Stronger properties

- Intermediately automorph-conjugate subgroup
- Sylow subgroup
- Hall subgroup
- Join of intermediately automorph-conjugate subgroups

### Weaker properties

## 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