# Join of finitely many subnormal subgroups

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 |

## Definition

A subgroup $H$ of a group $G$ is termed a join of finitely many subnormal subgroups if there exist finitely many subnormal subgroups $H_1, H_2, \dots, H_n$ of $G$ such that $H$ is the join $\langle H_1, H_2, \dots, H_n \rangle$.

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

## 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 condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

Suppose $H \le K \le G$ are groups such that $H$ is the join of finitely many subnormal subgroups in $G$. Then, $H$ is the join of finitely many subnormal subgroups in $K$.