Help:Subgroup property lookup

This page describes how to look up a certain kind of definition article

This page tries to give a description that'll help one locate the definition of a subgroup properties as well as facts and proofs relating to it.

Determining whether the term at hand is a subgroup property

Logical criterion

Logically, a subgroup property is something that, given any group and any subgroup, is either true of false for the subgroup in the group. That is, a given subgroup either has the property inside the group, or it does not have the property.

This criterion can be used to judge whether the term being defined is a subgroup property, by actually asking this question. However, the criterion is useless if we have absolutely no idea of either the meaning or the context.

Verbal criterion

Verbally, a subgroup property could be expressed in:

• Adjective form: This is typically written as the adjective followed by the word subgroup. For instance, the subgroup property of being a normal subgroup has the word normal (an adjective) followed by the word subgroup. Similarly for characteristic subgroup, verbal subgroup, permutable subgroup and so on.

• Adverb-qualified adjective form: This is typically written as the adverb followed by the adjective followed by the word subgroup. The adverb may, in some cases, be operating as a subgroup property modifier, usually, though it is simply an indicator of variation. Examples are fully characteristic subgroup, hereditarily normal subgroup, potentially characteristic subgroup.

• Initials form: These are subgroup properties that are best expressed in a whole sentence, but for ease, we abbreviate and write the initials followed by the word subgroup. For instance, the term NE-subgroup is used for a subgroup that equals the intersection of its normalizer and its normal closure. Similarly, the term CEP-subgroup denotes subgroups that have the congruence extension property (abbreviated as CEP).

• Member noun form: Property names like retract, direct factor and central factor.

Pitfalls

Usually it is true that any term obtained by prefixing an adjective to the word subgroup describes a subgroup property. However, this need to always be true.

Some typical pitfalls are:

• The term at hand may actually describe a group property being evaluated or measured for the group embedded as a subgroup. For instance, the term cyclic subgroup is actually the group property of being cyclic being evaluated on the subgroup.

In this case, the property can be viewed as a subgroup property, but the right place to look for and study it would be as a group property. For this purpose, simply replace the word subgroup by group and hunt for that group property.

• The term at hand may be a property of subgroups that can be evaluated in groups with some additional structure. For instance, we may talk of closed subgroup, discrete subgroup, and so on, which make sense in a topological group. These are not actually subgroup properties but rather properties as subsets in a topological space.

The right thing to do here is to identify the correct additional structure and then look for the property space of subgroups relative to that additional structure.

• The term at hand may represent a variation on the concept of subgroup. For instance, twisted subgroup and near subgroup are variations (weakenings) on the notion of subgroup.

General format of a subgroup property article

An article on a subgroup property will follow the property definition article format with some special things for subgroup properties. Further details are provided below.

Figuring out the definition of a subgroup property

Further information: Groupprops:Definition

Symbol-free definition

The symbol-free definition typically begins as follows: "A subgroup of a group is said to (have the given property) if ...." It is in the last part that the true conditions to be checked for satisfying the property are described.

There are the following typical formats to a symbol-free definition:

• As several equivalent definitions: Here each of the equivalent definitions is given as separate bullet points
• As many conditions that need to be simultaneously satisfied: Here, the various conditions are listed as bullet points, and it is required that a subgroup fulfil all of the condition

Sometimes an entire scenario needs to be developed to describe the conditions that need to be checked for the subgroup property. In this case, a symbol-free definition may not be appropriate.

Definition with symbols

The definition with symbols typically begins as follows: "A subgroup $H$ of a group $G$ is said to (have the given property) if ..." The main difference from the symbol-free definition is that the symbols $H$ and $G$ as well as many other temporary symbols may be used in this form of the definition.

Importance of understanding both forms of the definition

The definition with symbols is particularly useful when there are lots of cross-references, since the ambiguity of pronouns is avoided. On the other hand, there are situations where the symbol-free definition brings out the meaning more crisply and clearly. Moreover, the symbol-free definition may be more amenable to manipulation as there isn't an extra baggage of symbols to lug around. (The extra baggage of symbols could get particularly confusing if the same symbol is used for multiple purposes).

Equivalence of definitions

The fact that different definitions of a subgroup property are actually equivalent is usually proved in a separate definition equivalence page.

Formalisms

For some terms, there may be simpler definitions obtained using property-theoretic terms. For instance, there may be a simpler definition that observes that the given property is obtained by applying a certain subgroup property modifier to another subgroup property, or that it can be characterized in terms of a suitable formalism.

The definition in property-theoretic terms or formalistic terms is usually given in a separate section titled Formalisms.

Other facts about the property

The general format for subgroup properties has:

• A relation with other properties section where weaker and stronger properties are listed and also where other variations, opposites etc. are provided.
• Information on whether it satisfies a given metaproperty.

More information along the lines of determining whether a subgroup property satisfies some metaproperties is available at Subgroup metaproperty satisfaction:lookup.

Narrowing down an elusive property (reverse search)

This actually considers the reverse question -- suppose I have a definition in mind for a subgroup property. How do I determine whether a subgroup property having that definition has been studied so far?

Normal closure of the subgroup is the whole group

Suppose the property for which I want to find a name is characterized as follows: "the normal closure of the subgroup in the group is the whole group. Here are many different ways to attack this name-finding problem:

• Observe that this property is in some sense opposite to the property of being normal -- because the property of being normal can be characterized as the property of the normal closure being the subgroup itself. Hence, the right place to look at is Category:Opposites of normality. This category has a lot of possible candidates, such as abnormal subgroup, malnormal subgroup and contranormal subgroup, and one needs to check each of them to locate which one is the property one seeks.

• Observe that this property has something to do with the normal closure. So the right place to check out is the normal closure page. Since what we're looking for is a subgroup property related to the normal closure, the right place to look at is the related subgroup properties section. In this section, we easily spot the property we're looking for -- the property of being contranormal.

Subgroup has finite index in its normal closure

Consider the following subgroup property: the subgroup has finite index in its normal closure. Here are some approaches to determining the subgroup property:

• Since this property is closely related to normality (in fact, it is satisfied for every normal subgroup) one appropriate place to look for it is Category:Variations of normality. This category is too big, and searching in the entire category could be painful. Instead, let's also observe that the given subgroup property is always true for every subgroup of a finite group. Hence, it comes in Category:Subgroup properties tautological for finite groups. We easily see that there are only two subgroup properties in the intersection of these two categores: almost normal subgroup and nearly normal subgroup.

• Another way to locate the subgroup property is to try doing a search for "normal closure" and "finite index" and we get two entries: nearly normal subgroup and maximal subgroup. Clearly, maximal subgroup is not what we're looking for, so we're left with nearly normal subgroup.