# Difference between revisions of "Ascendant subgroup"

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==Definition== | ==Definition== | ||

− | + | A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''ascendant''' if we have subgroups <math>H_\alpha</math> of <math>G</math> for every ordinal <math>\alpha</math> such that: | |

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− | A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''ascendant''' if we have | ||

* <math>H_0 = H</math> | * <math>H_0 = H</math> | ||

− | * <math>H_\alpha \ \underline{\triangleleft} \ H_{\alpha + 1}</math> (viz <math>H_\alpha</math> is a [[normal subgroup]] of <math>H_{\alpha + 1}</math>) | + | * <math>H_\alpha \ \underline{\triangleleft} \ H_{\alpha + 1}</math> (viz <math>H_\alpha</math> is a [[normal subgroup]] of <math>H_{\alpha + 1}</math>) for every ordinal <math>\alpha</math>. |

− | * If <math>\alpha</math> is a limit ordinal, then <math>H_\alpha = \langle H_\gamma \rangle_{\gamma < \alpha}</math>, i.e., it is the [[join of subgroups|join]] of all preceding subgroups. | + | * If <math>\alpha</math> is a limit ordinal, then <math>H_\alpha = \bigcup_{\gamma < \alpha} H_\gamma</math>, i.e., it is the union of all preceding subgroups. Note that the union of any ascending chain of subgroups is indeed a sugbroup (in fact, more generally, [[directed union of subgroups is subgroup]]). We can also define <math>H_{\alpha}</math>as <math>\langle H_\gamma \rangle_{\gamma < \alpha}</math>, i.e., it is the [[join of subgroups|join]] of all preceding subgroups. |

and such that there is some ordinal <math>\beta</math> such that <math>H_\beta = G</math>. | and such that there is some ordinal <math>\beta</math> such that <math>H_\beta = G</math>. | ||

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===Stronger properties=== | ===Stronger properties=== | ||

− | + | {| class="sortable" border="1" | |

− | + | ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |

− | + | |- | |

+ | | [[Weaker than::normal subgroup]] || we can get a series that reaches the group in one step. || || || | ||

+ | |- | ||

+ | | [[Weaker than::subnormal subgroup]] || we can get a series that reaches the group in finitely many steps. || (obvious) || [[ascendant not implies subnormal]] || {{intermediate notions short|ascendant subgroup|subnormal subgroup}} | ||

+ | |- | ||

+ | | [[Weaker than::hypernormalized subgroup]] || we can use the series where the subgroup for each successor ordinal is the normalizer in the whole group of the subgroup for the ordinal. || (obvious) || || {{intermediate notions short|ascendant subgroup|hypernormalized subgroup}} | ||

+ | |} | ||

===Weaker properties=== | ===Weaker properties=== | ||

− | + | {| class="sortable" border="1" | |

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

+ | |- | ||

+ | | [[Stronger than::serial subgroup]] || || || || || | ||

+ | |} | ||

===Opposites=== | ===Opposites=== | ||

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==Metaproperties== | ==Metaproperties== | ||

− | { | + | {| class="sortable" border="1" |

− | + | ! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols | |

− | + | |- | |

− | + | | [[satisfies metaproperty::transitive subgroup property]] || Yes || [[ascendance is transitive]] || If <math>H \le K \le G</math> are groups such that <math>H</math> is an ascendant subgroup of <math>K</math> and <math>K</math> is an ascendant subgroup of <math>G</math>, then <math>H</math> is an ascendant subgroup of <math>G</math>. | |

− | + | |- | |

− | + | | [[satisfies metaproperty::trim subgroup property]] || Yes || || Every group is ascendant in itself, and the trivial subgroup is ascendant in any group. | |

− | + | |- | |

− | + | | [[satisfies metaproperty::intermediate subgroup condition]] || Yes || [[ascendance satisfies intermediate subgroup condition]] || If <math>H \le K \le G</math> are groups such that <math>H</math> is an ascendant subgroup of <math>G</math>, then <math>H</math> is an ascendant subgroup of <math>K</math>. | |

− | + | |- | |

− | + | | [[satisfies metaproprty::strongly intersection-closed subgroup property]] || Yes || [[ascendance is strongly intersection-closed]] || If <math>H_i, i \in I</math>, are all ascendant subgroups of a group <math>G</math>, so is <math>\bigcap_{i \in I} H_i</math>. | |

− | + | |- | |

− | + | | [[satisfies metaproperty::transfer condition]] || Yes || [[ascendance satisfies transfer condition]] || If <math>H,K \le G</math> are subgroups and <math>H</math> is ascendant in <math>G</math>, then <math>H \cap K</math> is ascendant in <math>K</math>. | |

− | + | |- | |

− | + | | [[satisfies metaproperty::image condition]] || Yes || [[ascendance satisfies image condition]] || Suppose <math>H</math> is an ascendant subgroup of a group <math>G</math>, and <math>\varphi:G \to K</math> is a surjective homomorphism of groups. Then, <math>\varphi(H)</math> is an ascendant subgroup of <math>K</math>. | |

− | + | |} |

## Latest revision as of 17:47, 21 December 2012

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 |

## Contents

## Definition

A subgroup of a group is termed **ascendant** if we have subgroups of for every ordinal such that:

- (viz is a normal subgroup of ) for every ordinal .
- If is a limit ordinal, then , i.e., it is the union of all preceding subgroups. Note that the union of any ascending chain of subgroups is indeed a sugbroup (in fact, more generally, directed union of subgroups is subgroup). We can also define as , i.e., it is the join of all preceding subgroups.

and such that there is some ordinal such that .

### In terms of the ascendant closure operator

The subgroup property of being an ascendant subgroup is obtained by applying the ascendant closure operator to the subgroup property of being normal.

## Relation with other properties

### Stronger properties

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

normal subgroup | we can get a series that reaches the group in one step. | |||

subnormal subgroup | we can get a series that reaches the group in finitely many steps. | (obvious) | ascendant not implies subnormal | |FULL LIST, MORE INFO |

hypernormalized subgroup | we can use the series where the subgroup for each successor ordinal is the normalizer in the whole group of the subgroup for the ordinal. | (obvious) | |FULL LIST, MORE INFO |

### Weaker properties

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

serial subgroup |

### Opposites

## Metaproperties

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

transitive subgroup property | Yes | ascendance is transitive | If are groups such that is an ascendant subgroup of and is an ascendant subgroup of , then is an ascendant subgroup of . |

trim subgroup property | Yes | Every group is ascendant in itself, and the trivial subgroup is ascendant in any group. | |

intermediate subgroup condition | Yes | ascendance satisfies intermediate subgroup condition | If are groups such that is an ascendant subgroup of , then is an ascendant subgroup of . |

strongly intersection-closed subgroup property | Yes | ascendance is strongly intersection-closed | If , are all ascendant subgroups of a group , so is . |

transfer condition | Yes | ascendance satisfies transfer condition | If are subgroups and is ascendant in , then is ascendant in . |

image condition | Yes | ascendance satisfies image condition | Suppose is an ascendant subgroup of a group , and is a surjective homomorphism of groups. Then, is an ascendant subgroup of . |