Upward-closed normal subgroup

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 $$H \le K \le G$$, $$K \triangleleft G$$.
 * $$H \triangleleft G$$, 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.

Stronger properties

 * Abelian-quotient subgroup

Weaker properties

 * Normal subgroup

Metaproperties
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.

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, viz every group is upward-closed normal as a subgroup of itself.