# Left-transitively fixed-depth subnormal subgroup

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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]

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

## Contents

## Definition

A subgroup of a group is termed **left-transitively fixed-depth subnormal** in if there exists a natural number such that is left-transitively -subnormal in . In other words, whenever is a -subnormal subgroup of a group , is also -subnormal in .

Note that any subgroup that is left-transitively -subnormal is also left-transitively -subnormal for .

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Characteristic subgroup | invariant under all automorphisms; for characteristic subgroups, we can set | Characteristic of normal implies normal | Left-transitively 2-subnormal subgroup|FULL LIST, MORE INFO | |

Left-transitively 2-subnormal subgroup | obtained by setting | |FULL LIST, MORE INFO | ||

Cofactorial automorphism-invariant subgroup | Left-transitively 2-subnormal subgroup|FULL LIST, MORE INFO | |||

Subgroup-cofactorial automorphism-invariant subgroup | Left-transitively 2-subnormal subgroup|FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Subnormal subgroup | Normal not implies left-transitively fixed-depth subnormal | |FULL LIST, MORE INFO |

### Related properties

Property | Meaning | Proof of one non-implication | Proof of other non-implication | Notions stronger than both | Notions weaker than both |
---|---|---|---|---|---|

Right-transitively fixed-depth subnormal subgroup | |FULL LIST, MORE INFO | Subnormal subgroup|FULL LIST, MORE INFO |

## 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

If are such that is left-transitively -subnormal in and is left-transitively -subnormal in , then is left-transitively -subnormal in .

### Intersection-closedness

This subgroup property is finite-intersection-closed; a finite (nonempty) intersection of subgroups with this property, also has this property

View a complete list of finite-intersection-closed subgroup properties

An intersection of finitely many such subgroups again has the property. In particular, the intersection of a left-transitively -subnormal subgroup and a left-transitively -subnormal subgroup is left-transitively -subnormal.

### 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

A join of finitely many such subgroups again has the property. In particular, the join of a left-transitively -subnormal subgroup and a left-transitively -subnormal subgroup is left-transitively -subnormal.