Category:Subgroup properties

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 group properties, sorted by importance rank:

Extremely important (importance rank one):

	Defining ingredient
Characteristic subgroup	Characteristic subalgebra
Direct factor	Internal direct product Normal subgroup Normal complement Retract
Improper subgroup
Maximal subgroup	Maximal operator Proper subgroup
Nontrivial subgroup	Trivial group
Normal subgroup	Homomorphism of groups Inner automorphism Conjugate subgroups Left coset of a subgroup Right coset of a subgroup Conjugacy class Commutator of two subgroups
Proper characteristic subgroup	Proper subgroup Characteristic subgroup
Proper normal subgroup	Proper subgroup Normal subgroup
Proper subgroup

Very important (importance rank two):

	Defining ingredient
Central factor	Inner automorphism Centralizer Normal subgroup Quotient group maps to outer automorphism group of normal subgroup Internal central product Image-potentially operator Direct factor Upper join-closure operator Cocentral subgroup
Central subgroup	Central factor Abelian group
Complemented normal subgroup
Fully invariant subgroup	Endomorphism of a group
Hall subgroup
Normal Hall subgroup	Normal subgroup Hall subgroup Subnormal subgroup Characteristic subgroup
Normal Sylow subgroup	Normal subgroup Sylow subgroup Finite group Subnormal subgroup Characteristic subgroup Fully invariant subgroup
Retract
Subnormal subgroup	Normal subgroup Subordination operator
Sylow subgroup	Group of prime power order Index of a subgroup Order of a group Hall subgroup

Somewhat important (importance rank three):

	Defining ingredient
Automorph-conjugate subgroup	Automorphic subgroups Conjugate subgroups Normal subgroup Normalizer of a subgroup
Permutable subgroup	Product of subgroups Permuting subgroups
Pronormal subgroup	Conjugate subgroups Join of subgroups

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 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

The page giving notations for important subgroup properties is:

Notation for subgroup properties