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.
We discuss four broad methods:
- Browse for links from terms to which the fact is related.
- Use semantic search to find the fact.
- Guess the name of the fact using our naming conventions.
- Use questions pages (these are available only for a very small subset of the site content).
We will illustrate each of these methods using the questions:
- "I've heard that a normal subgroup of a normal subgroup need not be normal. Is that true?"
- "Is a subgroup of a finitely generated group necessarily finitely generated?"
- Identity one or more terms to which the fact is related.
- Go to the page for the term and read it to find the link to the fact.
- In many cases, it is possible to identify a likely section within the page where the fact may appear.
Normal subgroups of normal subgroups
Consider the fact "a normal subgroup of a normal subgroup need not be normal" which you may have heard somebody say. You are interested in knowing whether the statement is true, as well as in a proof or an example that explains this.
The obvious initial place to look for this fact is the page on normal subgroup. This page is extremely long, so it would be impractical to read the page in its entirety. Instead, it makes sense to try to identify the appropriate section of the page that would cover an idea like this.
The first few sections (Definition, Importance, Examples) are clearly not relevant. The "Facts" section may appear the most relevant, but does not have the statement. This leaves the remaining sections, including "Metaproperties", "Relation with other properties", and "Effect of property operators."
If you are not familiar with the jargon used in the wiki, it may be hard to guess from the name which section is most relevant. However, eyeballing the sections reveals that "Metaproperties" is potentially most relevant, since it describes implications between multiple subgroup containments some of which may be normal. This means that this section is most likely to be worth focusing on. Having identified this section, it is easy to locate normality is not transitive as the relevant page.
Note that the "Effect of property operators" section also answers the question more indirectly.
Are subgroups of finitely generated groups finitely generated?
Let's say you have encountered the concept of finitely generated group, and you are wondering whether every subgroup of a finitely generated group is finitely generated. The obvious thing to do is to look at the page for finitely generated group. The Definition section doesn't have the information. The "Metaproperties" section looks like it might be relevant, and indeed, it answers the question in the negative: a subgroup of a finitely generated group need not be finitely generated. It also links to finite generation is not subgroup-closed which has a proof.
If, however, you miss that, there is a further section that contains the answer: the "Relation with other properties" section. This section defines Noetherian group as a group in which every subgroup is finitely generated, and links to the proof of the non-implication finitely generated not implies Noetherian. This is another way of putting the fact that a subgroup of a finitely generated group need not be finitely generated.
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.
Consider the statement:
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:
That is the name of the fact article.
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 is intersection-closed if an arbitrary nonempty intersection of subgroups with property also has property .
We can then reformulate the result as saying:
The idea is thus to abstract things into a metaproperty satisfaction.
Look at the result:
This can be thought of in terms of the composition operator that takes two subgroup properties and and outputs the property of being a subgroup such that there exists an intermediate subgroup such that satisfies in and satisfies in .
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:
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:
The principle behind property-theoretic organization is that a number of facts actually describe 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 normal 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.
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:
Clarifications and spill-overs
Some fact articles come as clarifications or spill-overs from other fact articles or definition articles, for instance:
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 is a normal subgroup of , and is an intermediate subgroup, then is also normal in . 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.