# Help:Property-theoretic lookup

The Group Properties Wiki is designed and organized using a property-theoretic paradigm. This article sketches briefly how to use the property-theoretic paradigm to look up facts in the wiki.

## So what's a property and what's the property-theoretic paradigm?

### What is a property?

A property over a collection of objects is something which classifies those objects into haves and have nots. In other words, given an object, it should either have that property or not have that property.

Examples from elementary mathematics:

• Being prime is a property over the collection of natural numbers. In other words, every natural number either has the property of being prime, or does not have the property of being prime.
• Being positive is a property over the collection of real numbers. In other words, every real number either has the property of being positive, or does not have the property of being positive.

Examples from group theory:

• Being solvable is a property over the collection of groups. In other words, every group either has the property of being solvable, or does not have the property of being solvable (every group is either solvable or not).
• Being normal is a property over the collection of subgroups. In other words, given a group and a subgroup of the group, the subgroup either has the property of being normal, or does not have the property of being normal.
• Being inner is a property over the collection of automorphisms. In other words, given a group and an automorphism of the group, the automorphism either has the property of being inner, or does not have the property of being inner.

Further information: Property

### How are properties related?

Given two properties over the same context space (i.e. two properties that are being used for the same collection of objects) we can ask questions like:

• Does the first property imply the second? For instance, is every odd number prime? (No).
• What do we get when we take the conjunction of the properties. A conjunction of two properties is the AND of the properties. For instance, the property of being an odd prime is the conjunction of the property of being odd and the property of being prime.

### Metaproperties: properties of properties

A metaproperty is a property that we can evaluate for properties over a certain context space. For instance:

• We can ask whether a property of natural numbers is additively closed. A property of natural numbers is additively closed if whenever $a,b$ satisfy the property, so does $a + b$. Thus, being additively closed is a metaproperty of natural numbers.
• We can ask whether a property of groups is subgroup-closed. A property of groups is termed subgroup-closed if any subgroup of a group with the property also has the property.
• We can ask whether a property of subgroups is transitive. A subgroup property $p$ is transitive if whenever $H \le K \le G$ are groups, such that $H$ satisfies $p$ in $K$ and $K$ satisfies $p$ in $G$, then $H$ satisfies $p$ in $G$.

### Property modifiers and operators

Starting with a property, we can apply a modifier or operator to it. For instance, given a group property, we can apply the hereditarily operator to it: this converts a property $p$ to the property of being a group in which every subgroup has property $p$.

A property modifier is a property operator that takes a property and outputs another property on the same context space (i.e. the same collection of objects). The hereditarily operator is a property modifier. There is a more general notion of property operator, which takes in inputs as properties (possibly multiple inputs from different property spaces) and outputs a property (possibly over a different property space). Here's one example. Given a group property $p$ and a subgroup property $q$, we can define the property of being a group in which every subgroup with subgroup property $q$, satisfies group property $p$ as an abstract group.

## Property theory for definition lookup

Most definitions related to properties come under the following headers. The categories for these headers are supercategories: they do not themselves list the properties, but rather, list the context spaces over which the properties are being evaluated.

Many of the individual categories have their own lookup pages.

## Property theory for fact lookup

Most facts related to properties come under the following headers. The categories for these headers are supercategories: they do not themselves list the properties, but rather, list the context spaces: