# Left-transitively 2-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]

## Definition

A subgroup of a group is termed a **left-transitively 2-subnormal subgroup** if it satisfies the following equivalent conditions:

- Whenever is a 2-subnormal subgroup of a group , is also a 2-subnormal subgroup of .
- Whenever is a normal subgroup of characteristic subgroup of a group , is also a normal subgroup of characteristic subgroup of .
- For any automorphism of , and any element , the automorphism of given as , where denotes conjugation by , preserves .

### Equivalence of definitions

`For full proof, refer: Equivalence of definitions of left-transitively 2-subnormal subgroup`

## Formalisms

### In terms of the left transiter

This property is obtained by applying the left transiter to the property: 2-subnormal subgroup

View other properties obtained by applying the left transiter

### In terms of the left transiter

This property is obtained by applying the left transiter to the property: normal subgroup of characteristic subgroup

View other properties obtained by applying the left transiter

## Relation with other properties

### Stronger properties

### Weaker properties

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

normal subgroup of characteristic subgroup | normal subgroup of a characteristic subgroup | Left-transitively 2-subnormal implies normal subgroup of characteristic subgroup | |FULL LIST, MORE INFO | |

2-subnormal subgroup | Normal subgroup of characteristic subgroup|FULL LIST, MORE INFO | |||

left-transitively fixed-depth subnormal subgroup | left-transitively -subnormal for some | |FULL LIST, MORE INFO |

### Incomparable properties

- Normal subgroup:
`For full proof, refer: Normal not implies left-transitively 2-subnormal, Left-transitively 2-subnormal not implies normal` - Right-transitively 2-subnormal subgroup

## Metaproperties

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)

Here is a summary:

Metaproperty name | Satisfied? | Proof | Difficulty level (0-5) | Statement with symbols |
---|---|---|---|---|

transitive subgroup property | Yes | left-transitive 2-subnormality is transitive | If are groups such that is left-transitively 2-subnormal in and is left-transitively 2-subnormal in , then is left-transitively 2-subnormal in . | |

trim subgroup property | Yes | Obvious reasons | 0 | For any group , and are characteristic in |

strongly intersection-closed subgroup property | Yes | left-transitive 2-subnormality is strongly intersection-closed | 1 | If , are all left-transitively 2-subnormal in , so is the intersection of subgroups . |

intermediate subgroup condition | No | left-transitive 2-subnormality does not satisfy intermediate subgroup condition | It is possible to have groups such that is left-transitively 2-subnormal in but not in . |