Large operator

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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:

Properties

Template:Reverse monotone pm

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.