# Groupprops:Organizational principles

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The main idea of the groupprops wiki is to organize the material available in group theory. For this purpose, certain organizational principles have been evolved and it is hoped that improved versions of these organizational principles will be adopted throughout the wiki.

This article describes the broad organizational principles. For specifics on how to go about writing, refer:

The article mainly focusses on organizational principles for the mathematical content. Organizational principles for the rest of the content (such as the historical background) shall be plugged in soon in a separate article.

## Property-theoretic approach for definition articles

### Identifying the property space

Further information: property space

The term solvable group refers to a property over the collection of groups, viz a group property. That is, given a group, we can ask the question: is this group solvable?

The term normal subgroup refers to a property over the collection of subgroups, viz a subgroup property. That is, given a subgroup of a group, we can ask the question: is this subgroup normal in the whole group?

The term inner automorphism refers to a property over the collection of automorphisms, viz an automorphism property. That is, given an automorphism of a group, we can ask the question: is this automorphism inner?

The idea that many of the terms that we define refer to properties over certain collections of objects is the beginning of an organizational paradigm that is in this wiki called property theory. The idea is to club all properties over a certain collection of objects into a construct called a property space, and then talk of operators on the property space.

The first organizational principle of this wiki is:

If the term being defined refers to a property, identify the property space. Then, while defining the term, use the template for that property space to define it.

For instance, whenever defining a subgroup property, we use the header template Template:Subgroup property and whenever defining a group property, we use the header template Template:Group property. The use of these templates does the following:

• It makes it very clear what the term being defined is, as in, whether it is a group property, a subgroup property, an automorphism property, a property of representations, a property of presentations, and so on. each template puts an emphasized line on top stating the kind of property that is being described, and also puts the article in the category corresponding to that property space. Thus, the article on normal subgroup begins with a line This article defines a subgroup property thanks to the use of Template:Subgroup property.
• It puts the property in the relevant category, and hence the property can be located in the category listing.
• In addition to the use of the template, we also get a structure for the overall article based on what the property is over. For instance, subgroup properties must satisfy a particular article format, which should cover issues such as the metaproperties they satisfy, the effect of applying operators on them, comparison with other properties, and so on. This is discussed later in the article.

A full listing of property space specification templates:

### Identifying metaproperties

Further information: metaproperty space

Rather than just have properties over objects themselves, we could have properties over properties, called metaproperties. A metaproperty over a property space can thus be evaluated (to true/false) for any given property in that property space. For instance, consider the notion of transitive subgroup property: a subgroup property $p$ is transitive if whenever $G$$H$$K$ such that $G$ satisfies property $p$ in $H$ and $H$ satisfies property $p$ in $K$, $G$ also satisfies property $p$ in $K$.

The term transitive here plays the role of a property, but it can evaluated only over something that is itself a subgroup property. Hence the notion of transitive subgroup property describes a subgroup metaproperty.

Thus, for each of the property spaces, we have a corresponding metaproperty space. Further, for each metaproperty space, we have a corresponding template. The subgroup metaproperty space uses the template Template:Subgroup metaproperty while the group metaproperty space uses the template Template: Group metaproperty.

A full listing of metaproperty space specification templates is available at:

Since all metaproperties are also properties, the metaproperty space specification templates are also listed as property space specification templates.

### Property modifiers and operators

There are a number of terms that describe operators on properties. For instance, the composition operator on subgroup properties takes as input two subgroup properties $p$ and $q$ and returns the property of being a subgroup $G$$K$ such that there is an intermediate subgroup $H$ where $G$ satisfies $p$ in $H$ and $H$ satisfies $q$ in $K$.

Such a thing is a binary subgroup property operator.

Similarly, there is a notion of left transiter, that takes as input a subgroup property and outputs a subgroup property. Since this is an operator from the subgroup property space to itself, it is termed a subgroup property modifier.

For each of these notions on each property space, we have templates. Thus, for a subgroup property modifier, the template is Template: Subgroup property modifier. As in the case of a property space specification template, there are two advantages:

• It makes it clear what kind of property operator we are talking about.
• It puts the operator in the relevant category.

A full listing of templates specifying the operator type is available at:

### Going further

Every term may not be a property, metaproperty, or property operator. There could be things like metmetaproperties, property operator properties, and so on. The crucial thing is to determine the kind of term being defined based on the property-theoretic way of describing and defining them.

For some terms, none of these property-theoretic ways may apply.

### Using the property-theoretic type to determine layout

The next important organizational principle is: The layout of the article depends on its property-theoretic type. Property definition articles are meant to follow the Property definition article format, and the general format is made more specific for a particular property space (where some of the things in the general foramt may not be applicable and where further things become applicable).

Thus, writing the article ,at least in part, involves filling out a form that depends on the property space. This ensures that the person writing the article does not miss out on a number of generic facts related to the term being defined.

Similarly, there are the:

## Organization of definition/terminology articles based on standardness

### Terminology local to the wiki

The wiki strongly encourages the use of terminology local to the wiki, so long as this terminology makes reasonable sense and does not conflict with standard terminology. However, terminology local to the wiki must be explicitly stated as being so, using Template:Wikilocal right at the top (after specifying the property space or otherwise the type).

### How basic is the term?

Another important parameter by which articles are classified is how basic they are. In this respect, we have the following:

A full list of terminology-level specification templates is available at:

## Relational organization

### Stronger and weaker

Within a given property space, there is a natural order among the properties via the implication relation. We say that property $p$ is stronger than $q$ or $p$$q$, or that $p$ implies $q$, if every element satisfying $p$ also satisfies $q$.

Thus, we would like the following:

• Given a property, we can find a listing of other properties which imply it (or are stronger than it). This may not be a complete listing but should at least provide a clue as to the properties that are reasonably close to it in nature and are still stronger than it.
• Given a property, we can find a listing of other properties which are implied by it (or are weaker than it). Again, the listing may not be complete but should cover close properties.
• Given two properties, we should be able to determine whether one of them implies the other, and if so, what the proof of the implication is.

The last of these actually gets covered under the later section on fact articles. For the listing of stronger and weaker properties, we have incorporated these into the property definition article format.

### Pivots, variation and opposites

The properties of being normal and pronormal may both be subgroup properties, but the first one (normality) is far more crucial and pivotal among subgroup properties while the latter is merely a variation on the theme of normality.

The notions of normal subgroup and contranormal subgroup are both subgroup properties and they represent intrinsically opposite flavors.

Thus, we have evolved the following concepts:

• The concept of variation on a property: This is not a rigorous notion, and is based more on a kind of loose understanding of the similarities between definitions. For instance, the property of being a potentially characteristic subgroup is a variation of the property of being a characteristic subgroup.

To label a property as a variation of another property, we use Template:Variationof, which also includes the property in a corresponding variational category. The variational category itself uses Template:Variationsof to indicate the pivot about which it has the variations. A listing of all variational categories is available at:

• The concept of opposites of a property: Again, very similar to the concept of variation. We label a property as an opposite of an existing property using Template:Oppositeof. This also puts it in an opposites category which further uses Template:Oppositesof. A listing of all opposites categories is available at:

### Applying metaproperties and property operators

This actually comes more under the header of fact articles, which is what we discuss next.

## Organization of fact articles

### The use of property theory in organizing fact articles

We view facts, not so much as historical incidents, but fundamentally as equations or relations between properties. This affects the way we name, state and prove facts. For instance, the fact that every characteristic subgroup of a normal subgroup is normal translates in terms of property theory to the statement that the result obtained by applying the composition operator for the properties of being characteristic and normal implies the property of being normal.

The name of the article itself reveals this property-theoretic thinking: Characteristic of normal implies normal.

The idea in fact articles is to:

• Provide definitions of the terms involved, which may include both the usual definition and the definition most amenable to interpreting and proving the given fact
• Analyze the steps of the proof
• Figure out what the related proofs are and what crucial techniques/ingredients are used in the proofs.

### Property implications

A very typical kind of fact of interest is that one property implies the other, that is, that every element satisfying a property $p$ must also satisfy the property $q$. Depending on the property space over which the particular implication holds, we have a corresponding template tag (and hence, also a category). For instance, characteristic implies normal is an implication of subgroup properties, and is hence tagged with the template Template:Subgroup property implication.

A full list of templates for property implications is available at:

### Operator computation

These are results which say something like: the result of applying so-and-so operator to this property gives that property. This comes under an operator computation template. A complete listing of operator computation templates is available at:

### Definition equivalence

Often, a definition article may give multiple definitions, the equivalence of all of the definitions not being direct or obvious. In this case, a separate article may be devoted to proving the equivalence of these definitions. Such an article is tagged using the Template:Definition equivalence template.

## Theme-based organization

Specifying the property space and the stronger and weaker properties doesn't always convey the complete flavour. For instance, the property of being a one-relator group and the property of being a semidirectly indecomposable group are both group properties but they arise from completely different aspects of group theory, the former stems from combinatorial group theory while the latter from the attempts to factorize a group.

Theme-based templates thus use historical, motivational and genetic themes to classify both definition articles and fact articles. The theme-based template both puts a message on the top and includes in a relevant category.

A full list of theme-based templates is available at: