Intersection-transiter

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

The intersection transiter operator is a map from the subgroup property space to the subgroup property space, that acts as transiter for the commutative associative binary property operator namely the intersection operator. The intersection transiter takes a subgroup property $p$ to the property of being a subgroup whose intersection with any subgroup having property $p$, also has property $p$.

Definition with symbols

Let $p$ be a subgroup property. The intersection transiter operator of $p$ is defined as the following property $q$: a subgroup $H$ has property $q$ in $G$ if and only if for every subgroup $K$ with property $p$ in $G$, $H\cap K$ also has property $p$ in $G$.