# Category:Group properties

From Groupprops

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):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.

### 2

### 3

### 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

### 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
- Group in which every 1-automorphism is automorphism class-preserving