| 2-subnormal subgroup | | Normal subgroup
Normal closure
Normalizer of a subgroup
Conjugate subgroups
Normal core of normalizer
Normal core
Subnormal subgroup
Subnormal depth
Commutator of two subgroups |
| 3-subnormal subgroup | | Normal subgroup
2-subnormal subgroup |
| 4-subnormal subgroup | | Subnormal subgroup
Subnormal depth
2-subnormal subgroup
Normal subgroup
3-subnormal subgroup |
| Abelian group | | Center
Derived subgroup
Normal subgroup
Diagonal-in-square operator |
| Abelian normal subgroup | | Normal subgroup
Abelian group |
| Abelian normal subgroup of maximum rank | | Abelian normal subgroup
Abelian group
Normal subgroup
Rank of a p-group
Normal rank of a p-group |
| Almost normal subgroup | | Normal subgroup
Subgroup of finite index
Normalizer of a subgroup
Index of a subgroup
Conjugate subgroups |
| Almost subnormal subgroup | | Normal subgroup
Subgroup of finite index |
| Automorph-conjugate subgroup | | Automorphic subgroups
Conjugate subgroups
Normal subgroup
Normalizer of a subgroup |
| Automorphism-faithful normal subgroup | | Automorphism-faithful subgroup
Normal subgroup |
| C-closed normal subgroup | | C-closed subgroup
Normal subgroup |
| Central factor | factor in central product
product with centralizer is whole group
quotient action by outer automorphisms is trivial
every inner automorphism restricts to an inner automorphism | 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 factor of normal subgroup | | Central factor
Normal subgroup
Normal closure |
| Centralizer-free normal subgroup | | Centralizer-free subgroup
Normal subgroup |
| Class two normal subgroup | | Normal subgroup
Group of nilpotency class two |
| Closed normal subgroup | | Closed subgroup of semitopological group
Normal subgroup |
| Commutator of a normal subgroup and a subset | | Normal subgroup
Commutator of two subsets
Commutator |
| Commutator-in-center subgroup | | Normal subgroup
Commutator-in-centralizer subgroup |
| Complete direct factor | | Direct factor
Complete group
Normal subgroup
Complemented normal subgroup |
| Complete group | | Normal subgroup
Direct factor |
| Completely divisibility-closed normal subgroup | | Completely divisibility-closed subgroup
Normal subgroup |
| Conjugacy-closed normal subgroup | | Conjugacy-closed subgroup
Normal subgroup |
| Coprime automorphism-faithful normal subgroup | | Coprime automorphism-faithful subgroup
Normal subgroup |
| Coprime automorphism-invariant normal subgroup | | Coprime automorphism-invariant subgroup
Normal subgroup |
| Coprime automorphism-invariant normal subgroup of group of prime power order | | Coprime automorphism-invariant normal subgroup
Group of prime power order
Normal subgroup
Normal subgroup of group of prime power order
Coprime automorphism-invariant subgroup
Coprime automorphism-invariant subgroup of group of prime power order |
| Cyclic normal subgroup | | Normal subgroup
Cyclic group |
| Dedekind group | every subgroup is normal
every cyclic subgroup is normal
normal closure of element is cyclic | Normal subgroup
Inner automorphism
Power automorphism
Normal closure
Levi operator
Cyclic group
Hamiltonian operator |
| Dedekind normal subgroup | | Normal subgroup
Dedekind group |
| Direct factor | factor in internal direct product
normal with normal complement
has centralizing complement | Internal direct product
Normal subgroup
Normal complement
Retract |
| Direct factor of normal subgroup | | Direct factor
Normal subgroup
Normal closure |
| Divisible normal subgroup | | Normal subgroup
Divisible group |
| Elementary abelian normal subgroup | | Normal subgroup
Elementary abelian group |
| Finite normal subgroup | | Normal subgroup
Finite group |
| Finitely generated normal subgroup | | Normal subgroup
Finitely generated group |
| Group in which every automorphism is inner | | Normal subgroup
Central factor
Automorphism of a group
Inner automorphism |
| Group in which every nontrivial normal subgroup has finite index | | Normal subgroup
Subgroup of finite index |
| Group in which every normal subgroup is a central factor | | Normal subgroup
Central factor |
| Group in which every normal subgroup is a direct factor | | Normal subgroup
Direct factor |
| Group in which every normal subgroup is characteristic | | Normal subgroup
Characteristic subgroup
Transitively normal subgroup
Automorphism of a group
Normal automorphism |
| Group in which every normal subgroup is fully invariant | | Normal subgroup
Fully invariant subgroup |
| Group in which every normal subgroup is powering-invariant | | Normal subgroup
Powering-invariant subgroup |
| Group in which every permutable subgroup is normal | | Permutable subgroup
Normal subgroup |
| Group in which every pronormal subgroup is normal | | Pronormal subgroup
Normal subgroup |
| Group in which every retract is a direct factor | | Normal subgroup
Retract
Direct factor |
| Group in which every subgroup is pronormal | | Pronormal subgroup
Normal subgroup
Hamiltonian operator |
| Group whose automorphism group is abelian | abelian automorphism group
any two automorphisms commute | Automorphism group of a group
Abelian group
Normal subgroup
Normal subgroup contained in centralizer of derived subgroup |
| Group whose inner automorphism group is central in automorphism group | | Inner automorphism group
Central subgroup
Automorphism group of a group
Inner automorphism
Central automorphism
Normal subgroup
Commutator-in-center subgroup |
| Groups embeddable as normal subgroups in a finite group with a common complement | | Normal subgroup
Permutable complements
Normal complement |
| Hall subgroup of normal subgroup | | Hall subgroup
Normal subgroup |
| Hereditarily normal subgroup | | Transitively normal subgroup
Dedekind group
Normal subgroup
Hereditarily operator |
