# Ascendant subgroup

From Groupprops

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 . |