Upward-closed subgroup property

Symbol-free definition
A subgroup property is said to be upward-closed or sometimes, right-hereditary, if whenever a subgroup has the property in a group, every intermediate subgroup also has the property in the group.

Definition with symbols
A subgroup property $$p$$ is termed upward-closed or sometimes right-hereditary if whenever $$G$$ &le; $$H$$ &le; $$K$$ are groups such that $$G$$ satisfies property $$p$$ in $$K$$, then $$H$$ also satisfies property $$p$$ in $$K$$.

In terms of the upward closure operator
A subgroup property is upward-closed if and only if it is unchanged under application of the upward closure operator. The upward closure operator is an idempotent property operator for subgroup properties.

Stronger metaproperties

 * Super-functional subgroup property

Weaker metaproperties

 * Join-closed subgroup property

Related metaproperties

 * Intermediate subgroup condition
 * Left-hereditary subgroup property
 * Right-hereditary subgroup property