# 2-subnormal subgroup

From Groupprops

## Definition

QUICK PHRASES: normal inside normal closure, every conjugate is in its normalizer, normal closure is in normalizer, normal subgroup of normal subgroup, subgroup of subnormal defect at most two

No. | Shorthand | A subgroup of a group is 2-subnormal in it if ... | A subgroup of a group if 2-subnormal in if ... |
---|---|---|---|

1 | Normal of normal | there is an intermediate subgroup containing it such that the subgroup is normal in the intermediate subgroup and such that the intermediate subgroup is normal in the whole group. | there is subgroup of such that is a normal subgroup of and is a normal subgroup of . |

2 | Normal in closure | the subgroup is normal in its normal closure in the whole group. | is a normal subgroup of its normal closure in . |

3 | Normal closure in normalizer | the normal closure of the subgroup is contained in the normalizer of the subgroup. | the normal closure is contained in the normalizer . |

4 | Every conjugate in normalizer | every conjugate of the subgroup is contained in its normalizer, i.e., every conjugate normalizes it. | for every , . |

5 | In normal core of normalizer | the subgroup is contained in its normal core of normalizer: the normal core of its normalizer. | is contained in the normal core of in . |

6 | Subnormal of depth 2 | it is a subnormal subgroup whose subnormal depth (also called subnormal defect) is at most . |
is subnormal in with subnormal depth at most . |

7 | contains second commutator | it contains its second commutator subgroup with the whole group | where denotes the commutator of two subgroups. |

This definition is presented using a tabular format. |View all pages with definitions in tabular format

This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definitionVIEW: Definitions built on this | Facts about this: (factscloselyrelated to 2-subnormal subgroup, all facts related to 2-subnormal subgroup) |Survey articles about this | Survey articles about definitions built on this

VIEW RELATED: Analogues of this | Variations of this | Opposites of this |

View a complete list of semi-basic definitions on this wiki

This page describes a subgroup property obtained as a composition of two fundamental subgroup properties: normal subgroup and normal subgroup

View other such compositions|View all subgroup properties

This is a variation of normality|Find other variations of normality | Read a survey article on varying normality

## Comment

A 2-subnormal subgroup has a unique fastest ascending subnormal series , where is the normal core of . It also has a unique fastest descending subnormal series , where is the normal closure of in . While subnormal subgroups of larger depth also have unique fastest descending subnormal series, they do not in general possess unique fastest ascending subnormal series. `Further information: 2-subnormal subgroup has a unique fastest ascending subnormal series, Subnormal subgroup has a unique fastest descending subnormal series, 3-subnormal subgroup need not have a unique fastest ascending subnormal series`

## Formalisms

### First-order description

This subgroup property is a first-order subgroup property, viz., it has a first-order description in the theory of groups.

View a complete list of first-order subgroup properties

A subgroup is 2-subnormal in a group if it satisfies the following first-order sentence:

## Examples

VIEW: subgroups of groups satisfying this property | subgroups of groups dissatisfying this propertyVIEW: Related subgroup property satisfactions | Related subgroup property dissatisfactions

## Relation with other properties

### Stronger properties

### Weaker properties

## Metaproperties

Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|

transitive subgroup property | No | 2-subnormality is not transitive | There exist groups , with 2-subnormal in , 2-subnormal in , but not 2-subnormal in . |

trim subgroup property | Yes | Every group is normal in itself, trivial subgroup is normal | For any group , the whole group and the trivial subgroup are both 2-subnormal. |

strongly intersection-closed subgroup property | Yes | Subnormality of fixed depth is strongly intersection-closed | all 2-subnormal subgroups of , then so is . |

finite-join-closed subgroup property | No | 2-subnormality is not finite-join-closed | Can have subgroups , both 2-subnormal in , such that is not 2-subnormal. |

conjugate-join-closed subgroup property | Yes | 2-subnormality is conjugate-join-closed | A join of subgroups of , with all 2-subnormal in and all conjugate to each other, is also 2-subnormal. |

intermediate subgroup condition | Yes | 2-subnormality satisfies intermediate subgroup condition | If , with 2-subnormal in , then is 2-subnormal in . |

transfer condition | Yes | 2-subnormality satisfies transfer condition | If , with 2-subnormal in , then is 2-subnormal in . |

image condition | Yes | 2-subnormality satisfies image condition | If 2-subnormal in , surjective homomorphism, then is 2-subnormal in . |

inverse image condition | Yes | 2-subnormality satisfies inverse image condition | If homomorphism, 2-subnormal in , then 2-subnormal in . |

upper join-closed subgroup property | No | 2-subnormality is not upper join-closed | Can have with 2-subnormal in both and but not in . |

## Effect of property operators

Operator | Meaning | Result of application | Proof |
---|---|---|---|

left transiter | if big group is 2-subnormal in a bigger group, so is subgroup | left-transitively 2-subnormal subgroup | by definition |

right transiter | any 2-subnormal subgroup of subgroup is 2-subnormal in whole group | right-transitively 2-subnormal subgroup | by definition |

join-transiter | join with any 2-subnormal subgroup is 2-subnormal | join-transitively 2-subnormal subgroup | by definition |