Category:Subgroup properties: Difference between revisions
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{{ | __NOTOC__ | ||
{{property category|subgroup}} | |||
Important subgroup properties, sorted by importance rank: | |||
{{ | Extremely important (importance rank one): {{#ask: [[Category:Subgroup properties]][[Importance rank::1]]|?defining ingredient|limit = 100|searchlabel = See more}} | ||
Very important (importance rank two): {{#ask: [[Category:Subgroup properties]][[Importance rank::2]]|?defining ingredient|limit = 100|searchlabel = See more}} | |||
Somewhat important (importance rank three): {{#ask: [[Category:Subgroup properties]][[Importance rank::3]]|?defining ingredient|limit = 100|searchlabel = See more}} | |||
==Related categories== | ==Related categories== | ||
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* [[:Category:Properties of subsets of groups]] | * [[:Category:Properties of subsets of groups]] | ||
* [[:Category:Subgroup pair properties]] | * [[:Category:Subgroup pair properties]] | ||
* [[:Category:Properties of subgroup series]] | |||
* [[:Category:Properties of subgroup families]] | |||
* [[:Category:Subgroup metaproperties]] | * [[:Category:Subgroup metaproperties]] | ||
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* [[:Category:Subgroup property testing problems]] | * [[:Category:Subgroup property testing problems]] | ||
===Facts and proofs related to subgroup properties=== | |||
* [[:Category:Subgroup property implications]] is a list of articles, each article describing an implication between two subgroup properties. For instance, [[characteristic implies normal]] meaning that every [[characteristic subgroup]] is a [[normal subgroup]]. | |||
* [[:Category:Subgroup property non-implications]] is a list of articles, each article describing why a particular subgroup property does ''not'' imply another one. Usually, the proof is constructive, giving an example of a subgroup satisfying the second property but not the first. For instance, [[normal not implies characteristic]] | |||
* [[:Category:Subgroup metaproperty satisfactions]] is a list of articles, each article describing how/why a given [[subgroup property]] satisfies a given [[subgroup metaproperty]]. | |||
* [[:Category:Subgroup metaproperty dissatisfactions]] is a list of articles, each article describing how/why a given [[subgroup property]] does ''not'' satisfy a given [[subgroup metaproperty]]. | |||
==Notations page== | ==Notations page== | ||
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The page giving notations for important subgroup properties is: | The page giving notations for important subgroup properties is: | ||
[[ | [[Notation for subgroup properties]] | ||
Latest revision as of 20:28, 7 December 2023
This is a category of properties for the following kind of objects: subgroups. In other words, this category lists subgroup properties
View a complete list of property categories
Important subgroup properties, sorted by importance rank:
Extremely important (importance rank one):
| Defining ingredient | |
|---|---|
| Characteristic subgroup | Characteristic subalgebra |
| Normal subgroup | Inner automorphism Conjugate subgroups Conjugacy class Commutator of two subgroups |
Very important (importance rank two):
| Defining ingredient | |
|---|---|
| Fully invariant subgroup | |
| Subnormal subgroup | Normal subgroup Subordination operator |
Somewhat important (importance rank three):
Related categories
Subcategory for pivotal elements
To see only the more important, or pivotal, among subgroup properties, refer Category:Pivotal subgroup properties.
Properties/metaproperties
Related categories include:
- Category:Properties of subsets of groups
- Category:Subgroup pair properties
- Category:Properties of subgroup series
- Category:Properties of subgroup families
- Category:Subgroup metaproperties
Testing/computational problems
- Category:Subgroup property implications is a list of articles, each article describing an implication between two subgroup properties. For instance, characteristic implies normal meaning that every characteristic subgroup is a normal subgroup.
- Category:Subgroup property non-implications is a list of articles, each article describing why a particular subgroup property does not imply another one. Usually, the proof is constructive, giving an example of a subgroup satisfying the second property but not the first. For instance, normal not implies characteristic
- Category:Subgroup metaproperty satisfactions is a list of articles, each article describing how/why a given subgroup property satisfies a given subgroup metaproperty.
- Category:Subgroup metaproperty dissatisfactions is a list of articles, each article describing how/why a given subgroup property does not satisfy a given subgroup metaproperty.
Notations page
The page giving notations for important subgroup properties is:
Subcategories
This category has the following 7 subcategories, out of 7 total.
C
Pages in category "Subgroup properties"
The following 200 pages are in this category, out of 728 total. The count includes redirect pages that have been included in the category. Redirect pages are shown in italics.
(previous page) (next page)A
- Abelian characteristic subgroup
- Abelian conjugacy-closed subgroup
- Abelian critical subgroup
- Abelian direct factor
- Abelian fully invariant subgroup
- Abelian hereditarily normal subgroup
- Abelian marginal subgroup
- Abelian normal subgroup
- Abelian normal subgroup of group of prime power order
- Abelian normal subgroup of maximum order
- Abelian pronormal subgroup
- Abelian subgroup
- Abelian subgroup of maximum order
- Abelian subgroup of maximum order which is normal
- Abelian subgroup of maximum rank
- Abelian subnormal subgroup
- Abelian Sylow subgroup
- Abelian verbal subgroup
- Abelian-completed subgroup
- Abelian-extensible automorphism-invariant subgroup
- Abelian-extensible endomorphism-invariant subgroup
- Abelian-potentially characteristic subgroup
- Abelian-potentially verbal subgroup
- Abelian-quotient subgroup
- Abelian-quotient-pullbackable automorphism-invariant subgroup
- Abnormal subgroup
- Action-isomorph-free subgroup
- AEP-subgroup
- Algebraic subgroup
- Almost normal subgroup
- Almost subnormal subgroup
- Amalgam-characteristic subgroup
- Amalgam-normal-subhomomorph-containing subgroup
- Amalgam-strictly characteristic subgroup
- Amalgamated free factor
- Ascendant subgroup
- Asymptotically fixed-depth join-transitively subnormal subgroup
- Autoclinism-invariant subgroup
- Automorph-commensurable subgroup
- Automorph-conjugate subgroup
- Automorph-dominating subgroup
- Automorph-join-closed subnormal subgroup
- Automorph-permutable subgroup
- Automorphically endomorph-dominating subgroup
- Automorphism-faithful AEP-subgroup
- Automorphism-faithful normal subgroup
- Automorphism-faithful subgroup
B
C
- C-closed normal subgroup
- C-closed self-centralizing subgroup
- C-closed subgroup
- C-normal subgroup
- C-subnormal subgroup
- Carter subgroup
- CDIN-subgroup
- Center of pronormal subgroup
- Center-fixing automorphism-balanced subgroup
- Center-fixing automorphism-invariant subgroup
- Central factor
- Central factor of normal subgroup
- Central factor of normalizer
- Central subgroup
- Central subgroup of finite group
- Central subgroup of normalizer
- Centralizer of involution
- Centralizer-annihilating endomorphism-invariant subgroup
- Centralizer-dense subgroup
- Centralizer-free normal subgroup
- Centralizer-free subgroup
- Centralizer-large subgroup
- Centrally closed subgroup
- Centrally large subgroup
- Characteristic AEP-subgroup
- Characteristic central factor
- Characteristic central subgroup
- Characteristic core of subgroup of finite index
- Characteristic direct factor
- Characteristic direct factor of abelian group
- Characteristic direct factor of nilpotent group
- Characteristic maximal subgroup of group of prime power order
- Characteristic subgroup
- Characteristic subgroup of abelian group
- Characteristic subgroup of center
- Characteristic subgroup of direct factor
- Characteristic subgroup of finite group
- Characteristic subgroup of finite index
- Characteristic subgroup of group of prime power order
- Characteristic subgroup of nilpotent group
- Characteristic subgroup of pronormal subgroup
- Characteristic subgroup of solvable group
- Characteristic subgroup of Sylow subgroup
- Characteristic transitively normal subgroup
- Characteristic-isomorph-free subgroup
- Characteristic-potentially characteristic subgroup
- Characteristically complemented characteristic subgroup
- Characteristically complemented normal subgroup
- Characteristically complemented subgroup
- Characteristically DP-decomposable subgroup
- Class two characteristic subgroup
- Class two normal subgroup
- Closure-characteristic subgroup
- Cocentral subgroup
- Cocentral subgroup of normalizer
- Cofactorial automorphism-invariant subgroup
- Commutator of a normal subgroup and a subset
- Commutator-in-center subgroup
- Commutator-in-centralizer subgroup
- Commutator-verbal subgroup
- Complemented central factor
- Complemented characteristic subgroup
- Complemented fully invariant subgroup
- Complemented homomorph-containing subgroup
- Complemented isomorph-containing subgroup
- Complemented normal subgroup
- Complemented transitively normal subgroup
- Complete characteristic subgroup
- Complete direct factor
- Completely divisibility-closed normal subgroup
- Completely divisibility-closed subgroup
- Component
- Composition subgroup
- Compressed subgroup
- Conjugacy-closed characteristic subgroup
- Conjugacy-closed Hall subgroup
- Conjugacy-closed normal subgroup
- Conjugacy-closed subgroup
- Conjugate-commensurable subgroup
- Conjugate-comparable subgroup
- Conjugate-dense subgroup
- Conjugate-join-closed subnormal subgroup
- Conjugate-large subgroup
- Conjugate-permutable subgroup
- Conjugation-invariantly permutably complemented subgroup
- Contracharacteristic subgroup
- Contranormal subgroup
- Coprime automorphism-faithful characteristic subgroup
- Coprime automorphism-faithful normal subgroup
- Coprime automorphism-faithful subgroup
- Coprime automorphism-invariant normal subgroup
- Coprime automorphism-invariant normal subgroup of group of prime power order
- Coprime automorphism-invariant subgroup
- Coprime automorphism-invariant subgroup of group of prime power order
- Core-characteristic subgroup
- Core-free 2-subnormal subgroup
- Core-free permutable subgroup
- Core-free subgroup
- Core-free subnormal subgroup
- Critical subgroup
- Cutting subgroup
- Cyclic characteristic subgroup
- Cyclic isomorph-containing subgroup
- Cyclic normal subgroup
- Cyclic normal subgroup of finite group
D
- Dedekind normal subgroup
- Descendant subgroup
- Diagonal subgroup of a direct power
- Direct factor
- Direct factor of characteristic subgroup
- Direct factor of complemented normal subgroup
- Direct factor of fully invariant subgroup
- Direct factor of normal subgroup
- Direct factor over central subgroup
- Direct power-closed characteristic subgroup
- Direct product of subgroups of factors in direct product decomposition into cyclic groups
- Direct projection-invariant subgroup
- Distinguished subgroup
- Divisibility-closed subgroup
- Divisibility-closed subgroup of abelian group
- Divisibility-closed subgroup of nilpotent group
- Divisible normal subgroup
- Double coset-ordering subgroup
- Double coset-separated subgroup
- DRC-subgroup
E
- EEP-subgroup
- Elementarily characteristic subgroup
- Elementary abelian normal subgroup
- Elementary abelian subgroup of maximum rank
- Elliptic subgroup
- Endoclinism-invariant subgroup
- Endomorph-dominating subgroup
- Endomorphism image
- Endomorphism kernel
- Epimarginal subgroup
- Existence-bound-word subgroup
- Existentially bound-word subgroup