# Join-transitively subnormal subgroup

## Definition

### Symbol-free definition

A subgroup of a group is termed **join-transitively subnormal** if its join (viz., the subgroup generated) with any subnormal subgroup is again subnormal.

### Definition with symbols

A subgroup of a group is termed **join-transitively subnormal** if whenever (viz., is subnormal in ), the join of subgroups is subnormal in .

BEWARE!This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

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 |

## Formalisms

### In terms of the join-transiter

This property is obtained by applying the join-transiter to the property: subnormal subgroup

View other properties obtained by applying the join-transiter

The subgroup property of being join-transitively subnormal is obtained by applying the join-transiter to the subgroup property of being subnormal.

## Relation with other properties

### Stronger properties

### Weaker properties

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

Subnormal subgroup | Finite-conjugate-join-closed subnormal subgroup|FULL LIST, MORE INFO | |||

Finite-automorph-join-closed subnormal subgroup | Join-transitively subnormal implies finite-automorph-join-closed subnormal | |FULL LIST, MORE INFO | ||

Finite-conjugate-join-closed subnormal subgroup | (via finite-automorph-join-closed subnormal subgroup) | Finite-automorph-join-closed subnormal subgroup|FULL LIST, MORE INFO |

## Metaproperties

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

Transitive subgroup property | No | normal implies join-transitively subnormal, subnormality is not finite-join-closed | We can have such that is join-transitively subnormal in and is join-transitively subnormal in , but is not join-transitively subnormal in . |

Trim subgroup property | Yes | Every group is join-transitively subnormal in itself; the trivial subgroup is join-transitively subnormal. | |

Finite-intersection-closed subgroup property | Known open problem | intersection problem for join-transitively subnormal subgroups | Given join-transitively subnormal subgroups of a group , is necessarily join-transitively subnormal? |

Intermediate subgroup condition | Possibly open problem (see intermediately join-transitively subnormal subgroup) | If such that is join-transitively subnormal in , is necessarily join-transitively subnormal in . | |

Finite-join-closed subgroup property | Yes | If are both join-transitively subnormal in , then is also join-transitively subnormal. |

### Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).

View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties

Clearly, the whole group is join-transitively subnormal, because its join with any subgroup is the whole group. Also, the trivial subgroup is join-transitively subnormal, because its join with any subnormal subgroup is the *same* subnormal subgroup.

### Join-closedness

YES:This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property.ABOUT THIS PROPERTY: View variations of this property that are join-closed | View variations of this property that are not join-closedABOUT JOIN-CLOSEDNESS: View all join-closed subgroup properties (or, strongly join-closed properties) | View all subgroup properties that are not join-closed | Read a survey article on proving join-closedness | Read a survey article on disproving join-closedness

By the general theory of transiters, the join-transiter of any subgroup property is itself a finite-join-closed subgroup property.