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]

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If the ambient group is a finite group, this property is equivalent to the property: subnormal subgroup
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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
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Relation with other properties

Stronger properties

• Subnormal subgroup

Weaker properties

• Join of subnormal subgroups

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.
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Suppose $H\leq K\leq G$ 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 $K$.