# Descendant 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 **descendant** if we have subgroups of for every ordinal such that:

- (i.e., is a normal subgroup of ) for every ordinal .
- If is a limit ordinal, then .

and such that there is some ordinal such that .

### In terms of the descendant closure operator

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

## Relation with other properties

### Stronger properties

### Weaker properties

### Related properties

### Opposites

## Facts

### Descendant-contranormal factorization

This result states that given any subgroup of , there is a unique subgroup containing such that is contranormal in and is descendant in .

## Metaproperties

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

transitive subgroup property | Yes | descendance is transitive | If are groups such that is a descendant subgroup of and is a descendant subgroup of , then is a descendant subgroup of . |

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

intermediate subgroup condition | Yes | descendance satisfies intermediate subgroup condition | If are groups such that is descendant in , then is descendant in . |

strongly intersection-closed subgroup property | Yes | descendance is strongly intersection-closed | If , are all descendant subgroups of , so is the intersection . |

image condition | No | descendance does not satisfy image condition | It is possible to have groups and , a descendant subgroup of and a surjective homomorphism such that is not a descendant subgroup of . |