Groupprops, The Group Properties Wiki (pre-alpha)

From Groupprops

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.

The complete list of subgroup properties is available at:

Category:Subgroup properties

Contents 1 Determining whether the term at hand is a subgroup property 1.1 Logical criterion 1.2 Verbal criterion 1.3 Pitfalls 2 =Narrowing down to smaller lists of subgroup properties 2.1 Pivotal subgroup properties 2.2 Subgroup properties subcategorized by metaproperties 2.3 Subgroup properties subcategorized by other criteria 3 Figuring out the definition of a subgroup property 3.1 Symbol-free definition 3.2 Definition with symbols 3.3 Importance of understanding both forms of the definition 3.4 Equivalence of definitions 3.5 Formalisms 4 Other facts about the property 4.1 The box on top 4.2 Examples 4.3 Relation with other properties 4.4 Metaproperties 5 Narrowing down an elusive property (reverse search) 5.1 Normal closure of the subgroup is the whole group 5.2 Subgroup has finite index in its normal closure

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.

In some cases, there may exist separate article on the group property viewed as a subgroup property; for instance, cyclic subgroup and Abelian subgroup.

• 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. For a complete list of such terms, refer Category:Variations of subgroup.

• The term at hand may represent a property that can be evaluated for subsets of groups, rather than for subgroups. The right place to look in this case is Category:Properties of subsets of groups.

=Narrowing down to smaller lists of subgroup properties

Pivotal subgroup properties

The category:

Category:Pivotal subgroup properties

lists only those subgroup properties that are pivotal or important. Thus, for instance, normal subgroup and characteristic subgroup are to be found in this list but malnormal subgroup is not. This list is extremely useful particularly for beginners who are likely to be interested only in the most important of subgroup properties.

Subgroup properties subcategorized by metaproperties

The category of all subgroup properties has subcategories based on metaproperties satisfied. For instance, if you're interested in subgroup properties that are closed under intersections, you can look in Category:Intersection-closed subgroup properties. If you're interested in subgroup properties satisfying the intermediate subgroup condition, you can look in Category:Subgroup properties satisfying intermediate subgroup condition.

This listing is available under Category:Subgroup properties. Also, the page for any subgroup property satisfying a metaproperty also links to the category (the list) of all subgroup properties satisfying that metaproperty.

Subgroup properties subcategorized by other criteria

• Category:Subgroup property conjunctions: This category lists subgroup properties obtained by taking the conjunction of two or more subgroup properties. For instance, self-centralizing normal subgroup and normal Hall subgroup.
• Category:Conjunctions of group property with subgroup property: This category lists subgroup properties obtained by taking the conjunction of a subgroup property and a group property. For instance, Abelian normal subgroup.
• Category:Subgroup properties tautological for finite groups: This category lists subgroup properties that are always true for subgroups of a finite group.

Figuring out the definition of a subgroup property

Further information: Ref:Subwiki: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. Some of the simplest symbol-free definitions are single-sentence definitions; for instance:

AN EXAMPLE OF A SINGLE-SENTENCE SYMBOL-FREE DEFINITION (Abelian normal subgroup): A subgroup of a group is termed an Abelian normal subgroup if it is Abelian as a group and normal as a subgroup.

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, or numbered points.

AN EXAMPLE OF A SYMBOL-FREE DEFINITION IN THE FORM OF SEVERAL EQUIVALENT DEFINITIONS (normal subgroup):
A subgroup of a group is said to be normal if it satisfies the following equivalent conditions:

1. It is the kernel of a :homomorphism from the group.
2. It is invariant under all inner automorphisms. Thus, normality is the invariance property with respect to the property of an automorphism being inner. This definition also motivates the term invariant subgroup for normal subgroup (which was used earlier).
3. It equals each of its conjugates in the whole group. This definition also motivates the term self-conjugate subgroup for normal subgroup (which was used earlier).
4. Its left cosets are the same as its right cosets (that is, it commutes with every element of the group).
5. It is a union of conjugacy classes.
6. It contains its commutator with the whole group.

• As many conditions that need to be simultaneously satisfied: Here, the various conditions are listed as bullet points or numbered points, and it is required that a subgroup fulfill all of the condition.

AN EXAMPLE OF A SYMBOL-FREE DEFINITION IN THE FORM OF MULTIPLE CONDITIONS (critical subgroup):
A subgroup of a group of prime power order is termed a critical subgroup if it is characteristic in the whole group and satisfies the following three conditions:

1. The subgroup is a Frattini-in-center group: Its Frattini subgroup is contained in its center.
2. The subgroup is a commutator-in-center subgroup: Its commutator with the whole group is contained in its center.
3. The subgroup is a self-centralizing subgroup: Its centralizer in the whole group is contained in it.

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.

TIP: When reading any definition that involves a list of conditions, take care to note whether the conditions listed are equivalent formulations of the definition, or whether they are separate conditions that need to be satisfied together.

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. As for the symbol-free definition, definitions with symbols may involve multiple equivalent formulations, and may also involve multiple conditions that need to be simultaneously satisfied.

AN EXAMPLE OF A DEFINITION WITH SYMBOLS IN THE FORM OF SEVERAL EQUIVALENT DEFINITIONS (normal subgroup)
A subgroup N of a group G is said to be normal in G (in symbols, or ^Notations) if the following equivalent conditions hold:

1. There is a homomorphism φ from G to another group such that the kernel of φ is precisely N.
2. For all g in G, . More explicitly, for all , we have .
3. For all g in G, gNg ^{− 1} = N.
4. For all g in G, gN = Ng.
5. N is a union of conjugacy classes.
6. The commutator [N,G] is contained in N.

AN EXAMPLE OF A DEFINITION WITH SYMBOLS IN THE FORM OF MULTIPLE CONDITIONS (critical subgroup):
Let G be a group of prime power order. A subgroup H of G is said to be critical if it is characteristic in G, and the following three conditions hold:

1. , i.e., the Frattini subgroup is contained inside the center (i.e., H is a Frattini-in-center group).
2. (i.e., H is a commutator-in-center subgroup of G).
3. C_G(H) = Z(H) (i.e., H is a self-centralizing subgroup of G).

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).

FOR MORE INFORMATION: If you want to learn more about the general philosophy regarding definitions and definition articles followed on subject wikis, refer Ref:Subwiki:Definition article and Ref:Subwiki:Definition.

Equivalence of definitions

The fact that different definitions of a subgroup property are actually equivalent is usually proved in a separate definition equivalence page. A link to this page is provided in a subsection of the definition section titled Equivalence of definitions. When there are many different definitions, different parts of the equivalence may be proved in different pages, or may require different forms of machinery.

AN EXAMPLE OF A DEFINITION EQUIVALENCE SECTION (normal subgroup):

• For the equivalence between definitions (1) and (2), refer Normal subgroup equals kernel of homomorphism.
• The equivalence between definitions (2) and (3) follows from a more general fact: Restriction of automorphism to subgroup invariant under it and its inverse is automorphism, combined with the fact that the inverse of an inner automorphism is also an inner automorphism (in fact, the inverse of conjugation by g is conjugation by g ^{− 1}). Further information: group acts as automorphisms by conjugation
• The equivalence between definitions (3) and (4) is a direct manipulation of equations involving elements and subsets. Further information: manipulating equations in groups
• The equivalence of definitions (2) or (3) with definition (5) is a straightforward unraveling of the meaning of conjugacy class.
• The equivalence of definitions (2) or (3) with definition (6) involves a straightforward unraveling of the meaning of commutator, along with a little bit of manipulation. Further information: manipulating equations in groups

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. This section usually follows the definition section (in pages where there is an Examples section, it may come after the examples section. The expression using any given formalism is accompanied by a brief description as well as links to information on the formalism as well as to other subgroup properties expressible using that formalism.

Other facts about the property

The box on top

Every subgroup property has a quotation box (a blockquote) on top stating that it is a subgroup property. This box provides links to the full list of subgroup properties. In addition, there are links to viewing related fact pages (these links are displayed only when fact pages do exist). There are six kinds of links shown:

• Subgroup property implications: These are statements asserting that one subgroup property implying the other, where the property we're looking at is somehow involved in the implication. Note that this listing has some overlap with the content in the Relation with other properties section. However, this list is automatically generated, giving it both advantages and disadvantages.
• Subgroup property non-implications: These are statements asserting that one subgroup property does not imply the other, where the property we're looking at is involved in the implication. Note that this listing has some overlap with the content in the Relation with other properties section. However, this list is automatically generated, giving it both advantages and disadvantages.
• Subgroup metaproperty satisfactions: These are statements asserting that the subgroup property satisfies a particular subgroup metaproperty (for instance, it may be transitive, or satisfy the intermediate subgroup condition). This listing has some overlap with the content in the Metaproperties section. However, this list is automatically generated, giving it both advantages and disadvantages.
• Subgroup metaproperty dissatisfactions: These are statements asserting that the subgroup property does not satisfy a particular subgroup metaproperty (for instance, it may be transitive, or satisfy the intermediate subgroup condition). This listing has some overlap with the content in the Metaproperties section. However, this list is automatically generated, giving it both advantages and disadvantages.
• Subgroup property satisfactions: These are statements asserting that a particular subgroup satisfies the property in a particular group. This may have some overlap with the Examples section, when such a section exists.
• Subgroup property dissatisfactions: These are statements asserting that a particular subgroup does not satisfy the property in a particular group. This may have some overlap with the Examples section, when such a section exists.

Examples

Some subgroup property pages have a section titled Examples. This section lists extreme examples, high-occurrence examples, low-occurrence examples, and other related material. The Examples section marks only the beginning of understanding examples. More examples can be obtained by looking at the Stronger properties section and spotting some of the non-implications: these give examples of cases where the subgroup satisfies the property but fails to satisfy a particular stronger property. Another source of examples is the Metaproperties section, where there are links to pages proving that the subgroup property does not satisfy a given metaproperty: most such proofs hinge on examples.

Relation with other properties

Most subgroup property pages have a section titled Relation with other properties. This section has the following typical subsections:

• Stronger properties: Lists important and closely related subgroup properties that are stronger than the given subgroup property. Links to pages proving the implication (as well as to pages proving the strictness of the implication) are provided when such pages exist. Sometimes, links to comparison survey articles comparing the two properties are also provided. The list need not be comprehensive, and a link to a more complete automatically generated list may be provided in some cases.
• Weaker properties: Lists important and closely related subgroup properties that are weaker than the given subgroup property. Links to pages proving the implication (as well as to pages proving the strictness of the implication) are provided when such pages exist. Sometimes, links to comparison survey articles comparing the two properties are also provided. The list need not be comprehensive, and a link to a more complete automatically generated list may be provided in some cases.
• Conjunction with other properties: Lists subgroup properties obtained by taking a conjunction (i.e., AND) of the given subgroup property and another subgroup property. This subsection may also include links to conjunctions with group properties.
• Incomparable/related properties: This subsection lists other subgroup properties that are neither stronger nor weaker, but are closely related. Links to proofs that neither implies the other may be given.

This section is thus a good way to get started with understanding the relation between the given subgroup property and other subgroup properties. There are, however, may other forms of relation that are not captured simply as implications. Further information: Help:Subgroup property implication lookup

Metaproperties

Most subgroup property pages have a section called metaproperties. This section has separate subsections for important metaproperties that the subgroup property either does or does not satisfy. A metaproperty is a property that can be evaluated for properties.

For instance, a normal subgroup of a normal subgroup need not be normal. We abstract this by saying that a transitive subgroup property is a subgroup property p such that whenever , with H satisfying p in K and K satisfying p in G, then H satisfies p in G. Thus, the fact that a normal subgroup of a normal subgroup is not normal is expressed as: normality is not transitive. This information is found in the Transitivity subsection on normality, where a link to a page with the full proof is also provided.

Note that the subsections do not provide a comprehensive list of all metaproperties that the given subgroup property satisfies. Further, a subsection on a metaproperty does not mean that the subgroup property does satisfy the metaproperty. For instance, the page on normal subgroup has a subsection titled Transitivity even though normality is not transitive. Further, links to proof pages are provided when such pages exist.

This section is a good way of understanding the way the subgroup property behaves, but it is only the beginning of such an understanding. Further information: Help: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.