# 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 propertyView 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 , the large operator applied to the property gives the following subgroup property . A subgroup of satisfes property in if given any subgroup satisfying in :

is trivial 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 says something like: *for every subgroup with property *. Thus, the more subgroups there are with property , the harder it is to be -large. More formally if , then -large -large.