Upward-closure operator

This article defines a subgroup property modifier (a unary subgroup property operator) -- viz an operator that takes as input a subgroup property and outputs a subgroup property

View a complete list of subgroup property modifiers OR View a list of all subgroup property operators (possibly with multiple inputs)

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Definition

Symbol-free definition

Let $p$ be a subgroup property. Then the upward closure of $p$ is defined as the property of being a subgroup such that all subgroups containing it satisfy property $p$ in the whole group.

Definition with symbols

Let $p$ be a subgroup property. Then, the upward closure of $p$ is defined as the following subgroup property $q$: A subgroup $H \le G$ satisfies property $q$ in $G$, if for every subgroup $K$ with $H \le K \le G$, $K$ satisfies $p$ in $G$.

Properties

Applying the upward closure operator twice is the same as applying it once. In other words, the properties that are fixed under the upward closure operator are precisely the same as the properties that can be obtained as images of the upward closure operator. A property that is fixed under the upward closure operator is termed an upward-closed subgroup property.

If $p \le q$ (both are subgroup properties) then the $UC(p) \le UC(q)$ where $UC$ denotes the upward closure.