# Help:Subgroup property exploration

The starting point to explore subgroup properties (if there is nothing particular you are looking for) are the category page listing all subgroup properties as well as the page defining subgroup property.

## Random exploration

Click on any subgroup property that seems interesting. Then, read about it. From each page, click randomly on a link to another subgroup property that seems interesting (such links could be found in the Relation with other properties section). Keep reading and drawing charts of the subgroup properties.

Random exploration could prove a very effective strategy.

## Using survey articles on subgroup properties

### The general idea

Like any collection of real-world objects, the subgroup properties are not all equally important -- some of them have a knack for appearing very often, in a lot of definitions and situations, while others appear very rarely. To get a feel for the important ones, check out Category: Pivotal subgroup properties (this one needs to be implemented).

You can then look at survey articles related to the pivotal subgroup properties. For instance, Category:Survey articles related to normality contains a list of articles that describe various aspects of the subgroup property of normality.

### Variational articles

Having read what a normal subgroup is, a nice place to get a feel for the subgroup properties closely related to normality, is the article varying normality. This article links to many notions closely related to normality, such as the notion of a permutable subgroup (which is a subgroup that permutes with every subgroup rather than with every element).

### Articles on importance/ubiquity

For many of the pivotal subgroup properties, there are separate articles that describe the importance/ubiquity of these subgroup properties. For instance, importance of normality describes why normality is important as a subgroup property, and why normal subgroups occur not just in pure group theory, but also in many applications to topological groups, group actions, Lie groups, algebraic groups, and more general notions of algebras than groups.

## Metaproperty-based exploration and property operator-based exploration

### Exploration based on metaproperties

Here, the focus is on subgroup properties that satisfy a particular subgroup metaproperty. For instance, suppose you start with the subgroup property of being a characteristic subgroup. There, you observe a subsection called Transitivity which states that the property of being a characteristic subgroup is transitive. That is, the property of being characteristic has the property of being transitive. You then read up the meaning of transitive subgroup property. The page on transitive subgroup property not only defines what a transitive subgroup property is, but also gives an idea of which subgroup properties are transitive and why.

Further, you can see a complete listing of transitive subgroup properties in Category:Transitive subgroup properties.

Similarly, the page on normality says that the property of normality is intersection-closed. This could lead you to read up about the metaproperty of being intersection-closed, and further, also see Category:Intersection-closed subgroup properties for a more complete listing.

The listing of subgroup metaproperties is to be found at Category:Subgroup metaproperties.

### Further exploration leading to property modifiers

After seeing that characteristicity is transitive but normality, for instance, is not, you may ask the question: What happens if a subgroup property is not transitive? The page on transitive subgroup property talks of three notions: the left transiter, the right transiter and the subordination. Each of these is a subgroup property modifier, viz it takes as input a subgroup property and outputs a subgroup property modifier. Further, the subgroup property output in each case is transitive.

Thus, we can explore the questions: if a given subgroup property is not transitive, what are its left transiter, right transiter, and subordination?

In the newly formatted articles on subgroup properties, this information is usually found in the section Property operators under the given articles. For instance, for the subgroup property of normality, it is found at Normal subgroup#Property operators.

Apart from left transiter, right transiter, and subordination, there are many other subgroup property modifiers. There are also a number of binary subgroup property operators, such as the composition operator, the intersection operator, and the meet operator.

The listing of subgroup property operators is at Category:Subgroup property operators.

## Comparing subgroup properties

### Find out which is stronger and which is weaker

We say that a subgroup property $p$ is stronger than a subgroup property $q$ if every subgroup of a group that has property $p$ also has property $q$.

To determine which subgroup property is stronger and which is weaker, you can try the following:

• In the article on the subgroup property $p$, look under the section Relation with other properties, and the subsections Stronger properties and Weaker properties. If the implication relation between $p$ and $q$ is important, then $q$ will be listed in the correct one among these. Sometimes, there is also a subsection titled Incomparable properties.
• There may be a separate page titled $p$ implies $q$ (or $q$ implies $p$, whichever is true) if the relation is important enough. This page will give the full statement and proof.

### Get a general flavor of the relation

For important closely related pairs of subgroup properties, there may be survey articles comparing them. For instance, normal versus characteristic compares the subgroup property of normality and characteristicity.

## Formalism-based exploration

### Finding information in the subgroup property article

The Groupprops wiki emphasizes the various formalisms that can be used to describe subgroup properties. In the newly formatted articles on subgroup property, there is a separate section titled Formalisms where all the expressions of the subgroup property in standard formalisms are described. For instance, in the article on normality, it is found at Normal subgroup#Formalisms.

For some of the incompletely/older formatted pages, information on formalisms may be found in the Definition section.

### More on the formalisms

The articles on formalisms for subgroup properties are still being reorganized and cleaned up! Please check back in a while.

## Theme-based exploration of subgroup properties

Chances are that you came to explore subgroup properties with particular interest in a certain kind of group theory. For instance, you may be interested in the theory of subgroups from the viewpoint of the lattice of subgroups of a group. You may be interested in studying subgroups from the viewpoint of Sylow theory. Or you may be interested in the theory of subgroups of finite index in infinite groups.

Unfortunately:

• Currently the subgroup properties (and even the overall Groupprops wiki) lacks good theme-based organization. Efforts in the direction of theme-based orgnaization have been started.
• Even whatever organization exists right now, is not easy to use for searching/locating relevant material