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 | Normal subgroup Derived subgroup Center Diagonal-in-square operator |
| Cyclic group | |
| Finite group |
Very important (importance rank two):
Somewhat important (importance rank three):
For the full list, including many not-so-important group properties, scroll to the bottom of this page.
Related categories
Related categories include:
Testing/computational problems
- 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.
Subcategories
This category has the following 7 subcategories, out of 7 total.
G
Pages in category "Group properties"
The following 200 pages are in this category, out of 528 total. The count includes redirect pages that have been included in the category. Redirect pages are shown in italics.
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A
- A-group
- Abelian group
- Abelian group of prime power order
- Abelian group that is finitely generated as a module over the ring of integers localized at a set of primes
- Abelian p-group
- Absolutely regular p-group
- Absolutely simple group
- Algebra group
- Algebraically closed group
- Almost quasisimple group
- Almost simple group
- Almost solvable group
- Ambivalent group
- Amenable discrete group
- Artinian group
- Automatic group
B
C
- C-group
- C-group (generating set sense)
- C1-group
- CA-group
- Camina group
- Capable group
- CCP-group
- Centerless group
- Centerless group that is upward-closed normal in its automorphism group
- Centralizer-connected group
- Centrally indecomposable group
- Characteristically Hopfian group
- Characteristically metacyclic group
- Characteristically metacylic group
- Characteristically polycyclic group
- Characteristically simple group
- CIA-group
- CN-group
- Co-Hopfian group
- Combable group
- Commutator-realizable group
- Complete group
- Complete solvable group
- Composition factor-permutable group
- Composition factor-unique group
- Composition series-unique group
- Con-Cos group
- Conjugacy-separable group
- Core-free group
- Countable group
- Countably free group
- Coxeter group
- CS-Baer Lie group
- CSCFN-realizable group
- Cyclic group
- Cyclic-center group
- Cyclotomic group
D
- Decomposable group
- Dedekind group
- Dicyclic group
- Dihedral group
- Direct product of finite groups
- Direct product of SmallGroup(16,3) and Z2
- Direct product of SmallGroup(16,4) and Z2
- Directed union-closed group property
- Directly indecomposable group
- Discriminating group
- Divisible group
- Divisible nilpotent group
E
F
- FC-group
- Finitary symmetric group
- Finite abelian group
- Finite ACIC-group
- Finite CN-group
- Finite complete group
- Finite cyclic group
- Finite elementary abelian group
- Finite group
- Finite group admitting a bijective quasihomomorphism to an abelian group
- Finite group having a subgroup series with prime indexes
- Finite group having the same orbit sizes of conjugacy classes and irreducible representations under automorphism group
- Finite group in which all cumulative conjugacy class size statistics values divide the order of the group
- Finite group in which all cumulative order statistics values divide the order of the group
- Finite group in which all partial sum values of squares of degrees of irreducible representations divide the order of the group
- Finite group in which every endomorphism is trivial or an automorphism
- Finite group in which half or more of the elements are involutions
- Finite group that is 1-isomorphic to an abelian group
- Finite group that is not isoclinic to a group of smaller order
- Finite group that is order statistics-equivalent to an abelian group
- Finite group whose automorphism group has equivalent actions on the sets of conjugacy classes and irreducible representations
- Finite group whose outer automorphism group is cyclic
- Finite group with cyclic quotient of automorphism group by class-preserving automorphism group
- Finite group with periodic cohomology
- Finite Lazard Lie group
- Finite nilpotent group
- Finite p-group in which the number of nth roots is a power of p for all n
- Finite p-group that is not characteristic in any finite p-group properly containing it
- Finite quasisimple group
- Finite rational group
- Finite Shmidt group
- Finite simple group
- Finite solvable group
- Finite supersolvable group
- Finite-(Frattini-embedded normal)-realizable group
- Finite-Frattini-realizable group
- Finitely generated abelian group
- Finitely generated conjugacy-separable group
- Finitely generated free abelian group
- Finitely generated free group
- Finitely generated FZ-group
- Finitely generated group
- Finitely generated group for which all homomorphisms to any finite group can be listed in finite time
- Finitely generated Hopfian group
- Finitely generated nilpotent group
- Finitely generated parafree group
- Finitely generated periodic group
- Finitely generated profinite group
- Finitely generated residually finite group
- Finitely generated simple group
- Finitely generated solvable group
- Finitely presented conjugacy-separable group
- Finitely presented group
- Finitely presented periodic group
- Finitely presented residually finite group
- Finitely presented simple group
- Finitely presented solvable group
- Finitely related group
- Fitting group
- Fitting-free group
- Fixed-class capable group
- Frattini-embedded normal-realizable group
- Frattini-free group
- Frattini-in-center group
- Free group
- Frobenius group
- FZ-group
G
- Gamma group
- Gamma-1-2 group
- Garside group
- Gaschütz group
- Generalized Camina group
- Global Lazard Lie group
- Global LCS-Lazard Lie group
- Global LUCS-Lazard Lie group
- Global UCS-Lazard Lie group
- Gradable nilpotent group
- Group admitting a nontrivial partition
- Group admitting a partition into abelian subgroups
- Group admitting a partition into cyclic subgroups
- Group embeddable in a finitary symmetric group
- Group generated by abelian normal subgroups
- Group generated by finitely many periodic elements
- Group generated by involutions
- Group generated by nilpotent pronormal subgroups
- Group generated by periodic elements
- Group having a class-inverting automorphism
- Group having a Sylow tower
- Group having an abelian conjugate-dense subgroup
- Group having an abelian contranormal subgroup
- Group having an abelian normal non-central subgroup
- Group having an automorphism whose restriction to the center is the inverse map
- Group having no free non-abelian subgroup
- Group having no proper cocentral subgroup
- Group having no proper isoclinic subgroup
- Group having no proper subgroup of finite index
- Group having subgroups of all orders dividing the group order
- Group in which all elements of finite order have a common bound on order
- Group in which all minimal generating sets have the same size
- Group in which all subnormal subgroups have a common bound on subnormal depth
- Group in which any two characteristic subgroups are comparable
- Group in which any two elements generating the same cyclic subgroup are automorphic
- Group in which any two normal subgroups are comparable
- Group in which class equals maximum subnormal depth
- Group in which elements of coprime finite orders commute