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

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.

Metaproperty-based exploration and property operator-based exploration

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.

You then ask the question: What happens if a subgroup property is not transitive? You explore the ideas of left transiter and right transiter and start studying proofs of what is the left transiter of what. You have then moved into the realm of property operator-based exploration.

Implication/variation based exploration

This involves the study of subgroup properties that are closely related to one another. For instance, suppose you start with the subgroup property of being normal. You then happen to see, under Relation with other properties the category on variations of normality. Clicking on that, you come across the notion of permutable subgroup. Then, you look at subgroup properties stronger and weaker than permutable subgroup. Among the weaker properties, you pick on conjugate-permutable subgroup. Then, you come across the fact that every conjugate-permutable subgroup is subnormal. You are prompted to read a proof of this fact.