Group properties:Lookup

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

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

This criterion can be used to judge whether the term being defined is a group 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 group property could be expressed in:


 * Adjective form: This is typically written as the adjective followed by the word group. For instance, the subgroup property of being a nilpotent group has the word nilpotent (an adjective) followed by the word group. Similarly for simple group, Abelian group, solvable group and so on.


 * Adverb-qualified adjective form: This is typically written as the adverb followed by the adjective followed by the word group. The adverb may, in some cases, be operating as a group property modifier, usually, though it is simply an indicator of variation. Examples are almost simple group, locally finite group, semidirectly indecomposable group.


 * Initials form: These are group properties that are best expressed in a whole sentence, but for ease, we abbreviate and write the initials followed by the word group. For instance, the term A-group is used for a finite group in which every Sylow subgroup is an Abelian group. Similarly, the terms PE-group, FC-group, FZ-group


 * Property noun form: Property names like Schreier property, subnormal intersection property.


 * Property name which is a full phrase: These names typically begin with group of or group with. The names are often self-descriptive. For instance group of prime power order, group of finite homological type.

Pitfalls
Usually it is true that any term obtained by prefixing an adjective to the word group describes a group property. However, this need not always be true.

Some typical pitfalls are:


 * The term at hand may actually be a group with additional structure or arising in a specific context. For instance, topological group is not a group that satisfies the property of being topological, rather it is a group with an additional compatible structure of a topological space. Similarly for Lie group, algebraic group, formal group.


 * The term at hand may be a property over groups with additional structure. For instance, compact group is not a group property, it is a property of topological groups.


 * The term at hand may be a variation on the concept of group.

The right thing to do here is to identify what the additional structure or variational structure is and search accordingly.

General format of a group 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.

Symbol-free definition
The symbol-free definition typically begins as follows: "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 group $$G$$ is said to (have the given property) if ..." The main difference from the symbol-free definition is that the symbol $$G$$ 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 group 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 group property modifier to another group property, or that it can be characterized in terms of a suitable formalism.

The definition in property-theoretic or formalistic terms is typically given in a 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 group property satisfies some metaproperties is available at group metaproperty satisfaction:lookup. Information about understanding which group properties imply which is given at group property implication:lookup.

Narrowing down an elusive property (reverse search)
This actually considers the reverse question -- suppose I have a definition in mind for a group property. How do I determine whether a group property having that definition has been studied so far?

Normal subgroups of normal subgroups are normal
Suppose the property for which I want to find a name is characterized as follows: "every normal subgroup of a normal subgroup of the group is normal".


 * One direction of attempt is as follows: observe that in general it is not true that a normal subgroup of a normal subgroup is normal. The long article on normality mentions this fact under the subsection Transitivity, where it links to the page normality is not transitive. That page, in turn, tells us that those groups in which normality is transitive, are termed T-groups.


 * Another approach is to observe that if a normal subgroup of a normal subgroup were indeed normal, then every subnormal subgroup is normal. This suggests we look at the relation between normal subgroups and subnormal subgroups, and gets us to the page subnormal not implies normal. The partial truth section here mentions T-groups.


 * Another approach is to simply read the article on subnormal subgroups, where the Related group properties section mentions T-groups.

Every finitely generated subgroup is finite
Suppose the property for which I want to find a name is: "every finitely generated subgroup is finite".


 * One approach is to first check out the article on finite group, since the property at hand is close to finiteness. There, there is a link to Category:Variations of finiteness (groups) and visiting this category gives a list of properties. Searching around among these properties gives the required property: locally finite group.