# 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

### Symbol-free definition

A subgroup of a group is said to be **ascendant** if there is an ascending series of subgroups indexed by ordinals, each normal in the next, that starts at the given subgroup, and terminates at the whole group.

### Definition with symbols

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

- (viz is a normal subgroup of )
- If is a limit ordinal, then , 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

### Weaker properties

### Opposites

## Metaproperties

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

transitive subgroup property | Yes | 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. |