Join of finitely many subnormal subgroups

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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

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 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.