Large 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)

Definition

Given a subgroup property $p$, the large operator applied to the property $p$ gives the following subgroup property $q$. A subgroup $H$ of $G$ satisfes property $q$ in $G$ if given any subgroup $K$ satisfying $p$ in $G$:

$H \cap K$ is trivial $\implies K$ is trivial

Application

Some important instances of application of the large operator:

• large subgroup: obtained from the tautology subgroup property
• normality-large subgroup: obtained from normal subgroup

Properties

Note that the property of being large with respect to $p$ says something like: for every subgroup with property $p$. Thus, the more subgroups there are with property $p$, the harder it is to be $p$-large. More formally if $p \le q$, then $q$-large $\implies p$-large.