# Join of finitely many subnormal subgroups

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 a **join of finitely many subnormal subgroups** if there exist finitely many subnormal subgroups of such that is the join .

Note that in a group satisfying subnormal join property, being a join of finitely many subnormal subgroups is precisely equivalent to being a subnormal subgroup.

## Formalisms

### In terms of the finite-join-closure

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

View other properties obtained by applying the finite-join-closure

## Relation with other properties

### Stronger properties

### Weaker properties

## Metaproperties

### Intermediate subgroup condition

YES:This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup conditionABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

Suppose are groups such that is the join of finitely many subnormal subgroups in . Then, is the join of finitely many subnormal subgroups in .