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From Groupprops

Contents 1 Finding facts 2 The naming of a fact article 2.1 Implications 2.2 Metaproperty satisfactions 2.3 Property operators/modifiers 3 Using categories to locate facts 3.1 Property-theoretic organization 3.2 Organization based on level and standardness 3.3 Theme-based organization 3.4 Clarifications and spill-overs 4 Links to facts from definition articles 5 Location by analogy 6 Having located the fact article

Finding facts

You can use groupprops to:

• Find exact statements of facts related to some area
• Find proofs of facts once you know their statements

Finding a fact is typically not as easy as finding a definition, because unlike terms to be defined, facts don't have names. The exceptions are famous named theorems, but even for these, you need to know the theorem name, and the theorem name cannot be naturally derived from the statement.

The naming of a fact article

Typically, the naming of any fact article is determined by what the fact says in simple property-theoretic language. For those facts that are named after their discoverers, or have other standard names, either that name is used instead of the property-theoretic name, or that name is redirected to the property-theoretic name. For instance, Feit-Thompson theorem redirects to odd-order implies solvable.

In certain cases, however, historical names are preferred because the statement is too complicated to admit a simple property-theoretic name. This is particularly true for more advanced and technical results, such as those occurring in the classification of finite simple groups.

Below we describe some approaches to property-theoretic naming.

Implications

Consider the statement:

Every characteristic subgroup is normal.

Here, the statement describes an implication from the subgroup property of being characteristic to the subgroup property of being normal. This fact can thus be written as:

Characteristic implies normal

That is the name of the fact article.

Metaproperty satisfactions

Consider the statement:

An arbitrary nonempty intersection of normal subgroups is normal.

To get an easy name for this statement, we think of the metaproperty of being intersection-closed. Basically, we say that:

A subgroup property p is intersection-closed if an arbitrary nonempty intersection of subgroups with property p also has property p.

We can then reformulate the result as saying:

Normality is intersection-closed

The idea is thus to abstract things into a metaproperty satisfaction.

Property operators/modifiers

Look at the result:

Every characteristic subgroup of a normal subgroup is normal.

This can be thought of in terms of the composition operator that takes two subgroup properties p and q and outputs the property of being a subgroup such that there exists an intermediate subgroup K such that H satisfies p in K and K satisfies q in G.

The above result then says that:

Appying the composition operator to the subgroup properties of being characteristic and normal givens something which implies the subgroup property of being normal.

We encode this as:

Characteristic of normal implies normal

Using categories to locate facts

As you can probably guess from the above, locating a fact article by directly guessing its name is often hard, and the category organization of facts is helpful in finding facts. We discuss here some ways this can be achieved. All fact articles can be found in one or more of the subcategories of the category:

Category:Facts

Property-theoretic organization

The principle behind property-theoretic organization is that a number of facts actually desribe relations between properties. For instance, some facts could be translated as saying that one property is stronger than another, some facts can be translated as saying that a certain property satisfies a certain metaproperty.

• Category:Property implications: The categories listed under this give instances of one property implying another. For instance, any characteristic subgroup is normal: this is an instance of one subgroup property implying another, and is listed in Category:Subgroup property implications.
• Category:Property non-implications: The categories listed under this give instances of one property not implying another. For instance, it is not true that any normal subgroup is characteristic, and the article for this is found in Category:Subgroup property non-implications.
• Category:Metaproperty satisfactions: This category lists instances of a property satisfying a metaproperty. Rather, it lists categories that list metaproperty satisfactions in various context spaces (for instance, group metaproperty satisfactions, subgroup metaproperty satisfactions). For instance, the statement an intersection of normal subgroups is normal is an example of the subgroup property called normality satisfying the subgroup metaproperty of being intersection-closed. The article name is normality is intersection-closed.
• Category:Metaproperty dissatisfactions: This category lists instances of a property not satisfying a metaproperty. The category itself has no articles, rather it lists categories for metaproperty dissatisfactions in various context spaces (like subgroup metaproperty dissatisfactions, group metaproperty dissatisfactions). For instance, the statement a normal subgroup of a normal subgroup need not be norma is an example of the subgroup property called normality not satisfying the subgroup metaproperty of being transitive. The article name is normality is not transitive.

Organization based on level and standardness

Some important categories for this are:

• Category:Basic facts in group theory: These are basic, standard facts related to group theory.
• Category:Semi-basic facts in group theory: These are standard facts, but they may either involve non-basic concepts or terminology, or have more involved proofs.
• Category:Elementary non-basic facts in group theory: These are elementary facts in group theory, but are not part of a basic theorem list. They are often observations/lemmas that are useful in certain specific contexts but are not widely used, even if the proofs ar every elementary. For instance, the fact that union of all conjugates is proper, or that product of conjugates is proper.

Theme-based organization

Some fact categories are related to specific themes, which may be disciplines of or related to group theory, or specific problems. Peruse the subcategory structure for the following categories:

• Category:Facts related to specific disciplines
• Category:Facts related to specific problems

Clarifications and spill-overs

Some fact articles come as clarifications or spill-overs from other fact articles or definition articles, for instance:

• They may arise as lemmas in the proof of other fact articles Fill this in later
• Category:Proofs of equivalence of definitions: They may be required to show the equivalence of multiple definitions.

Links to facts from definition articles

One of the best strategies to zero down on a fact article is to read the articles defining one or more of the term intricately linked with the fact. For example, suppose you want to find a proof of the fact that if H is a normal subgroup of G, and K is an intermediate subgroup, then H is also normal in K. This seems hard to search on directly; however, reading the article on normal subgroup yields that in the metaproperty section, there is a statement exactly of that sort. Further, the statement gives a link to the corresponding fact article: Normality satisfies intermediate subgroup condition.

The whole definition article might often be too long to locate the fact that one wants, so it is better to know which subsection to look in.

• Facts that describe why certain definitions or formulations of the definition are equivalent, are usually linked to from the Definition section or the Formalisms section.
• Facts that relate the existing term with other terms of the same kind (for instance, property implications, property non-implications, which relate two properties over the same context space) are often linked to in the Relation with other ... section.
• Facts that describe properties/metaproperties/metametaproperties that the given term satisfies, may be found in the section headed Properties, Metaproperties or Metametaproperties.
• Facts that describe what happens when certain operations or modifications are performed on the given term, may be found in sections like Effect of operators or Effect of modifiers.

Location by analogy

Another powerful tool to locate facts is by analogy. Suppose you are looking for a certain fact, but cannot think of how to find it; then try to remember other facts that:

• Have similar statements, or have one or more common components to them (for instance, they involve the same property, same metaproperty).
• Have similar proof methods
• Use the given fact, or are used by the given fact

Then, look at the sections titled Facts/Results used, Related facts/results, Applications, Corollaries to search for the fact you're looking for.

Having located the fact article

After locating the fact article, start reading it! Please note that not all fact articles currently contain proofs of the facts; this is because they have not yet been fully developed. If you encounter a fact article without proof and would like a proof included, but do not know a proof, leave a note in the talk page of that fact article (by clicking the discussion tab above the article).

To learn more about the content and structure of a fact article, refer Groupprops:Fact article.