# Right-transitively fixed-depth subnormal 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]

If the ambient group is a finite group, this property is equivalent to the property:subnormal subgroup

View other properties finitarily equivalent to subnormal subgroup | View other variations of subnormal subgroup |

This is a variation of subnormality|Find other variations of subnormality |

## Definition

### Definition with symbols

A subgroup of a group is termed **right-transitively fixed-depth subnormal** if there exists such that is right-transitively -subnormal in : whenever is a -subnormal subgroup of , is also -subnormal in .

Any right-transitively -subnormal subgroup is also right-transitively -subnormal for .

(Here, a -subnormal subgroup is a subnormal subgroup whose subnormal depth is at most ).

## Relation with other properties

### Stronger properties

- Transitively normal subgroup: Here, . Also related:
- Right-transitively 2-subnormal subgroup: Here, . Also related:
- Nilpotent subnormal subgroup
- Finite subnormal subgroup

### Weaker properties

### Related properties

## Metaproperties

### Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitiveABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

### Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).

View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties

### 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