Category:Group properties

This is a category of properties for the following kind of objects: groups. In other words, this category lists group properties
View a complete list of property categories

Important group properties, sorted by importance rank:

Extremely important (importance rank one):

	Defining ingredient
Abelian group	Center Derived subgroup Normal subgroup Diagonal-in-square operator
Cyclic group
Finite abelian group
Finite cyclic group	Group of integers modulo n
Finite group
Group of prime order
Symmetric group	Permutation

Very important (importance rank two):

	Defining ingredient
Finite nilpotent group
Finite solvable group	Finite group Solvable group Polycyclic group Composition series Chief series
Finitely generated abelian group	External direct product
Finitely generated group
Finitely presented group	Presentation of a group Finitely generated group
Free group
Group of nilpotency class two	Nilpotency class Derived subgroup Center Inner automorphism group Abelian group
Group of prime power order
Metabelian group	Solvable group Derived length Derived subgroup Abelian group Abelian normal subgroup Abelian-quotient subgroup Commutator Meta operator
Metacyclic group	Cyclic normal subgroup Cyclic group
Nilpotent group	Center Commutator of two subgroups Lower central series Subnormal subgroup Square of a group Commutator Diagonal-in-square operator
Simple group	Normal subgroup Surjective homomorphism of groups Homomorphism of groups Trivial homomorphism Subnormal subgroup Simple group operator
Solvable group	Normal series Abelian group Subnormal series Derived series Derived subgroup Characteristic series Fully invariant series
Supersolvable group

Somewhat important (importance rank three):

	Defining ingredient
Almost simple group
Ambivalent group	Conjugate elements Real element
Characteristically simple group	Characteristic subgroup Characteristic closure Characteristic core Simple group operator
Finitely generated nilpotent group
Hopfian group
Monolithic group	Minimal normal subgroup Normal subgroup
Noetherian group	Hereditarily operator Finitely generated group
Quasisimple group
Rational group	Fully normalized subgroup
Rationally powered group
Residually finite group	Subgroup of finite index Normal subgroup of finite index Profinite completion of a group Subdirect product Direct product of finite groups Profinite topology Residually operator Finite group
SQ-universal group	Finitely generated group Subquotient Finitely generated free group Free group:F2

For the full list, including many not-so-important group properties, scroll to the bottom of this page.

Related categories

Related categories include:

• Category:Subgroup properties

Testing/computational problems

• Category:Group property testing problems

Facts and proofs related to group properties

• Category:Group property implications is a page listing implication relations between pairs of group properties, viz describing how/why one group property must imply another.
• Category:Group property non-implications is a page listing non-implication relations between pairs of group properties, viz describing how/why one group property does not imply another.