Upward-closed normal subgroup

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This property of a normal subgroup is completely characterized by the abstract isomorphism class of the quotient group

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Definition

Symbol-free definition

A subgroup of a group is termed upward-closed normal if it satisfies the following equivalent conditions:

• Every subgroup intermediate between the given subgroup and the whole group, is normal in the whole group.
• It is normal and the quotient group is a Dedekind group, viz all subgroups of the quotient group are normal in it.

Definition with symbols

A subgroup $H$ of a group $G$ is termed upward-closed normal if it satisfies the following equivalent conditions:

• For any subgroup $K$ such that $H\le K\le G$, $K◃G$.
• $H◃G$, and the quotient group $G/H$ is a Dedekind group, viz., all its subgroups are normal.

In terms of the upward closure operator

The property of being an upward-closed normal subgroup is obtained by applying the upward closure operator to the subgroup property of being normal.

Relation with other properties

Stronger properties

• Abelian-quotient subgroup

Weaker properties

• Normal subgroup

Metaproperties

Transitivity

NO: This subgroup property is not transitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole group
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Clearly, an upward-closed normal subgroup of an upward-closed normal subgroup need not be normal. In fact, in any solvable non-Abelian group, the second derived subgroup is an upward-closed normal subgroup of an upward-closed normal subgroup, yet it is usually not itself upward-closed normal.

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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The property of being upward-closed normal satisfies the intermediate subgroup condition, viz if $H\le K\le G$ and $H$ is upward-closed normal in $G$, $H$ is also upward-closed normal in $K$. This follows essentially from the fact that normality satisfies the intermediate subgroup condition.

Trimness

The property of being upward-closed normal is not trivially true. However, it is an identity-true subgroup property, viz every group is upward-closed normal as a subgroup of itself.