Abelian group
type: Ontology
creation date: 2025-11-09T11:36:29+00:00
imports: http://semantic-mediawiki.org/swivt/1.0

Abelian group
type: Subject
type: Category:Basic definitions in group theory
type: Category:Group properties
type: Category:Pivotal group properties
type: Category:GAP-testable group properties
type: Category:Standard terminology
page: https://groupprops.subwiki.org/wiki/Abelian_group
wiki namespace: 0
wiki page content language: en
defining ingredient: Normal subgroup
defining ingredient: Derived subgroup
defining ingredient: Center
defining ingredient: Diagonal-in-square operator
importance rank: 1
MSC class: 20K
obtained by applying: Diagonal-in-square operator
page class: Term
satisfies metaproperty: Subgroup-closed group property
satisfies metaproperty: Quotient-closed group property
satisfies metaproperty: Varietal group property
satisfies metaproperty: Direct product-closed group property
stronger than: Nilpotent group
stronger than: Solvable group
stronger than: Metabelian group
stronger than: Virtually abelian group
stronger than: FZ-group
stronger than: FC-group
stronger than: Bell group
weaker than: Cyclic group
weaker than: Locally cyclic group
weaker than: Finitely generated abelian group
weaker than: Homocyclic group
weaker than: Residually cyclic group
weaker than: Epabelian group
weaker than: Finite abelian group
wiki page modification date: 2023-12-29T15:18:11Z
modification date#aux: 2460308.1376273
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http://groupprops.subwiki.org/wiki/Special:URIResolver/Abelian_group-23_QUERY1b9d56a7093975833c44a78db8058328
type: Subject
master page: Abelian group
wiki namespace: 0
query depth: 1
query format: list
query size: 4
query string: FZ-group Abelian group
wiki page sort key: Abelian group# QUERY1b9d56a7093975833c44a78db8058328

http://groupprops.subwiki.org/wiki/Special:URIResolver/Abelian_group-23_QUERY2debc9b2311e0c1c49c51a9cfc9488c4
type: Subject
master page: Abelian group
wiki namespace: 0
query depth: 1
query format: broadtable
query size: 4
query string: FZ-group Abelian group
wiki page sort key: Abelian group# QUERY2debc9b2311e0c1c49c51a9cfc9488c4

http://groupprops.subwiki.org/wiki/Special:URIResolver/Abelian_group-23_QUERY97990178055aa9b34c2f95fdf09a262a
type: Subject
master page: Abelian group
wiki namespace: 0
query depth: 1
query format: plainlist
query size: 4
query string: FC-group Abelian group
wiki page sort key: Abelian group# QUERY97990178055aa9b34c2f95fdf09a262a

http://groupprops.subwiki.org/wiki/Special:URIResolver/Abelian_group-23_QUERYfdf082505d07139637aaf24fc24e10af
type: Subject
master page: Abelian group
wiki namespace: 0
query depth: 1
query format: list
query size: 4
query string: FC-group Abelian group
wiki page sort key: Abelian group# QUERYfdf082505d07139637aaf24fc24e10af

http://groupprops.subwiki.org/wiki/Special:URIResolver/Abelian_group-23_QUERY79785d9324722cb8dae78efd5cf7b833
type: Subject
master page: Abelian group
wiki namespace: 0
query depth: 1
query format: broadtable
query size: 4
query string: FC-group Abelian group
wiki page sort key: Abelian group# QUERY79785d9324722cb8dae78efd5cf7b833

http://groupprops.subwiki.org/wiki/Special:URIResolver/Abelian_group-23_QUERYa9508cb25b0443f5d2d54cae1e251973
type: Subject
master page: Abelian group
wiki namespace: 0
query depth: 1
query format: plainlist
query size: 4
query string: Bell group Abelian group
wiki page sort key: Abelian group# QUERYa9508cb25b0443f5d2d54cae1e251973

http://groupprops.subwiki.org/wiki/Special:URIResolver/Abelian_group-23_QUERY424c6a5a970830657afda1c77d88aa87
type: Subject
master page: Abelian group
wiki namespace: 0
query depth: 1
query format: list
query size: 4
query string: Bell group Abelian group
wiki page sort key: Abelian group# QUERY424c6a5a970830657afda1c77d88aa87

http://groupprops.subwiki.org/wiki/Special:URIResolver/Abelian_group-23_QUERY920113fd15f4882b5a871584a0da8788
type: Subject
master page: Abelian group
wiki namespace: 0
query depth: 1
query format: broadtable
query size: 4
query string: Bell group Abelian group
wiki page sort key: Abelian group# QUERY920113fd15f4882b5a871584a0da8788

http://groupprops.subwiki.org/wiki/Special:URIResolver/Abelian_group-23_QUERYa1136ba471c50b97991562a8e87bef2a
type: Subject
master page: Abelian group
wiki namespace: 0
query depth: 1
query format: plainlist
query size: 2
query string: Diagonal-in-square operator
wiki page sort key: Abelian group# QUERYa1136ba471c50b97991562a8e87bef2a

Symmetric group:S3
type: Subject
page: https://groupprops.subwiki.org/wiki/Symmetric_group-3AS3
wiki namespace: 0
wiki page content language: en
dissatisfies property: Abelian group
wiki page sort key: Symmetric group:S3

Direct product of Z5 and S3
type: Subject
page: https://groupprops.subwiki.org/wiki/Direct_product_of_Z5_and_S3
wiki namespace: 0
wiki page content language: en
dissatisfies property: Abelian group
wiki page sort key: Direct product of Z5 and S3

Direct product of Z3 and D10
type: Subject
page: https://groupprops.subwiki.org/wiki/Direct_product_of_Z3_and_D10
wiki namespace: 0
wiki page content language: en
dissatisfies property: Abelian group
wiki page sort key: Direct product of Z3 and D10

Holomorph of Q8
type: Subject
page: https://groupprops.subwiki.org/wiki/Holomorph_of_Q8
wiki namespace: 0
wiki page content language: en
dissatisfies property: Abelian group
wiki page sort key: Holomorph of Q8

Quaternion group
type: Subject
page: https://groupprops.subwiki.org/wiki/Quaternion_group
wiki namespace: 0
wiki page content language: en
dissatisfies property: Abelian group
wiki page sort key: Quaternion group

Dihedral group:D14
type: Subject
page: https://groupprops.subwiki.org/wiki/Dihedral_group-3AD14
wiki namespace: 0
wiki page content language: en
dissatisfies property: Abelian group
wiki page sort key: Dihedral group:D14

Dihedral group:D22
type: Subject
page: https://groupprops.subwiki.org/wiki/Dihedral_group-3AD22
wiki namespace: 0
wiki page content language: en
dissatisfies property: Abelian group
wiki page sort key: Dihedral group:D22

Dihedral group:D26
type: Subject
page: https://groupprops.subwiki.org/wiki/Dihedral_group-3AD26
wiki namespace: 0
wiki page content language: en
dissatisfies property: Abelian group
wiki page sort key: Dihedral group:D26

Projective special linear group:PSL(3,2)
type: Subject
page: https://groupprops.subwiki.org/wiki/Projective_special_linear_group-3APSL-283-2C2-29
wiki namespace: 0
wiki page content language: en
dissatisfies property: Abelian group
wiki page sort key: Projective special linear group:PSL(3,2)

Alternating group:A4
type: Subject
page: https://groupprops.subwiki.org/wiki/Alternating_group-3AA4
wiki namespace: 0
wiki page content language: en
dissatisfies property: Abelian group
wiki page sort key: Alternating group:A4

Dihedral group:D8
type: Subject
page: https://groupprops.subwiki.org/wiki/Dihedral_group-3AD8
wiki namespace: 0
wiki page content language: en
dissatisfies property: Abelian group
wiki page sort key: Dihedral group:D8

General affine group:GA(1,11)
type: Subject
page: https://groupprops.subwiki.org/wiki/General_affine_group-3AGA-281-2C11-29
wiki namespace: 0
wiki page content language: en
dissatisfies property: Abelian group
wiki page sort key: General affine group:GA(1,11)

Alternating group:A6
type: Subject
page: https://groupprops.subwiki.org/wiki/Alternating_group-3AA6
wiki namespace: 0
wiki page content language: en
dissatisfies property: Abelian group
wiki page sort key: Alternating group:A6

Cyclic group:Z2
type: Subject
page: https://groupprops.subwiki.org/wiki/Cyclic_group-3AZ2
wiki namespace: 0
wiki page content language: en
satisfies property: Abelian group
wiki page sort key: Cyclic group:Z2

Cyclic group:Z3
type: Subject
page: https://groupprops.subwiki.org/wiki/Cyclic_group-3AZ3
wiki namespace: 0
wiki page content language: en
satisfies property: Abelian group
wiki page sort key: Cyclic group:Z3

Elementary abelian group:E25
type: Subject
page: https://groupprops.subwiki.org/wiki/Elementary_abelian_group-3AE25
wiki namespace: 0
wiki page content language: en
satisfies property: Abelian group
wiki page sort key: Elementary abelian group:E25

Cyclic group:Z30
type: Subject
page: https://groupprops.subwiki.org/wiki/Cyclic_group-3AZ30
wiki namespace: 0
wiki page content language: en
satisfies property: Abelian group
wiki page sort key: Cyclic group:Z30

Elementary abelian group:E49
type: Subject
page: https://groupprops.subwiki.org/wiki/Elementary_abelian_group-3AE49
wiki namespace: 0
wiki page content language: en
satisfies property: Abelian group
wiki page sort key: Elementary abelian group:E49

Cyclic group:Z71
type: Subject
page: https://groupprops.subwiki.org/wiki/Cyclic_group-3AZ71
wiki namespace: 0
wiki page content language: en
satisfies property: Abelian group
wiki page sort key: Cyclic group:Z71

Cyclic group:Z103
type: Subject
page: https://groupprops.subwiki.org/wiki/Cyclic_group-3AZ103
wiki namespace: 0
wiki page content language: en
satisfies property: Abelian group
wiki page sort key: Cyclic group:Z103

Cyclic group:Z107
type: Subject
page: https://groupprops.subwiki.org/wiki/Cyclic_group-3AZ107
wiki namespace: 0
wiki page content language: en
satisfies property: Abelian group
wiki page sort key: Cyclic group:Z107

Cyclic group:Z113
type: Subject
page: https://groupprops.subwiki.org/wiki/Cyclic_group-3AZ113
wiki namespace: 0
wiki page content language: en
satisfies property: Abelian group
wiki page sort key: Cyclic group:Z113

Cyclic group:Z61
type: Subject
page: https://groupprops.subwiki.org/wiki/Cyclic_group-3AZ61
wiki namespace: 0
wiki page content language: en
satisfies property: Abelian group
wiki page sort key: Cyclic group:Z61

Cyclic group:Z97
type: Subject
page: https://groupprops.subwiki.org/wiki/Cyclic_group-3AZ97
wiki namespace: 0
wiki page content language: en
satisfies property: Abelian group
wiki page sort key: Cyclic group:Z97

Cyclic group:Z73
type: Subject
page: https://groupprops.subwiki.org/wiki/Cyclic_group-3AZ73
wiki namespace: 0
wiki page content language: en
satisfies property: Abelian group
wiki page sort key: Cyclic group:Z73

Cyclic group:Z89
type: Subject
page: https://groupprops.subwiki.org/wiki/Cyclic_group-3AZ89
wiki namespace: 0
wiki page content language: en
satisfies property: Abelian group
wiki page sort key: Cyclic group:Z89

Cyclic group:Z11
type: Subject
page: https://groupprops.subwiki.org/wiki/Cyclic_group-3AZ11
wiki namespace: 0
wiki page content language: en
satisfies property: Abelian group
wiki page sort key: Cyclic group:Z11

Cyclic group:Z13
type: Subject
page: https://groupprops.subwiki.org/wiki/Cyclic_group-3AZ13
wiki namespace: 0
wiki page content language: en
satisfies property: Abelian group
wiki page sort key: Cyclic group:Z13

Cyclic group:Z17
type: Subject
page: https://groupprops.subwiki.org/wiki/Cyclic_group-3AZ17
wiki namespace: 0
wiki page content language: en
satisfies property: Abelian group
wiki page sort key: Cyclic group:Z17

Cyclic group:Z19
type: Subject
page: https://groupprops.subwiki.org/wiki/Cyclic_group-3AZ19
wiki namespace: 0
wiki page content language: en
satisfies property: Abelian group
wiki page sort key: Cyclic group:Z19

Cyclic group:Z23
type: Subject
page: https://groupprops.subwiki.org/wiki/Cyclic_group-3AZ23
wiki namespace: 0
wiki page content language: en
satisfies property: Abelian group
wiki page sort key: Cyclic group:Z23

Cyclic group:Z29
type: Subject
page: https://groupprops.subwiki.org/wiki/Cyclic_group-3AZ29
wiki namespace: 0
wiki page content language: en
satisfies property: Abelian group
wiki page sort key: Cyclic group:Z29

Cyclic group:Z31
type: Subject
page: https://groupprops.subwiki.org/wiki/Cyclic_group-3AZ31
wiki namespace: 0
wiki page content language: en
satisfies property: Abelian group
wiki page sort key: Cyclic group:Z31

Cyclic group:Z37
type: Subject
page: https://groupprops.subwiki.org/wiki/Cyclic_group-3AZ37
wiki namespace: 0
wiki page content language: en
satisfies property: Abelian group
wiki page sort key: Cyclic group:Z37

Cyclic group:Z41
type: Subject
page: https://groupprops.subwiki.org/wiki/Cyclic_group-3AZ41
wiki namespace: 0
wiki page content language: en
satisfies property: Abelian group
wiki page sort key: Cyclic group:Z41

Cyclic group:Z43
type: Subject
page: https://groupprops.subwiki.org/wiki/Cyclic_group-3AZ43
wiki namespace: 0
wiki page content language: en
satisfies property: Abelian group
wiki page sort key: Cyclic group:Z43

Cyclic group:Z47
type: Subject
page: https://groupprops.subwiki.org/wiki/Cyclic_group-3AZ47
wiki namespace: 0
wiki page content language: en
satisfies property: Abelian group
wiki page sort key: Cyclic group:Z47

Cyclic group:Z53
type: Subject
page: https://groupprops.subwiki.org/wiki/Cyclic_group-3AZ53
wiki namespace: 0
wiki page content language: en
satisfies property: Abelian group
wiki page sort key: Cyclic group:Z53

Cyclic group:Z59
type: Subject
page: https://groupprops.subwiki.org/wiki/Cyclic_group-3AZ59
wiki namespace: 0
wiki page content language: en
satisfies property: Abelian group
wiki page sort key: Cyclic group:Z59

Cyclic group:Z67
type: Subject
page: https://groupprops.subwiki.org/wiki/Cyclic_group-3AZ67
wiki namespace: 0
wiki page content language: en
satisfies property: Abelian group
wiki page sort key: Cyclic group:Z67

Cyclic group:Z79
type: Subject
page: https://groupprops.subwiki.org/wiki/Cyclic_group-3AZ79
wiki namespace: 0
wiki page content language: en
satisfies property: Abelian group
wiki page sort key: Cyclic group:Z79

Cyclic group:Z83
type: Subject
page: https://groupprops.subwiki.org/wiki/Cyclic_group-3AZ83
wiki namespace: 0
wiki page content language: en
satisfies property: Abelian group
wiki page sort key: Cyclic group:Z83

Cyclic group:Z21
type: Subject
page: https://groupprops.subwiki.org/wiki/Cyclic_group-3AZ21
wiki namespace: 0
wiki page content language: en
satisfies property: Abelian group
wiki page sort key: Cyclic group:Z21

Cyclic group:Z28
type: Subject
page: https://groupprops.subwiki.org/wiki/Cyclic_group-3AZ28
wiki namespace: 0
wiki page content language: en
satisfies property: Abelian group
wiki page sort key: Cyclic group:Z28

Direct product of Z2 and Z14
type: Subject
page: https://groupprops.subwiki.org/wiki/Direct_product_of_Z2_and_Z14
wiki namespace: 0
wiki page content language: en
satisfies property: Abelian group
wiki page sort key: Direct product of Z2 and Z14

Cube map is endomorphism iff abelian (if order is not a multiple of 3)
type: Subject
page: https://groupprops.subwiki.org/wiki/Cube_map_is_endomorphism_iff_abelian_-28if_order_is_not_a_multiple_of_3-29
wiki namespace: 0
wiki page content language: en
proves property satisfaction of: Abelian group
wiki page sort key: Cube map is endomorphism iff abelian (if order is not a multiple of 3)

Abelian IAPS
type: Subject
page: https://groupprops.subwiki.org/wiki/Abelian_IAPS
wiki namespace: 0
wiki page content language: en
analogue of: Abelian group
wiki page sort key: Abelian IAPS

Abelian normal subgroup
type: Subject
page: https://groupprops.subwiki.org/wiki/Abelian_normal_subgroup
wiki namespace: 0
wiki page content language: en
conjunction involving: Abelian group
wiki page sort key: Abelian normal subgroup
defining ingredient: Abelian group

Solvable group
type: Subject
page: https://groupprops.subwiki.org/wiki/Solvable_group
wiki namespace: 0
wiki page content language: en
defining ingredient: Abelian group
wiki page sort key: Solvable group
weaker than: Abelian group

2-Engel group
type: Subject
page: https://groupprops.subwiki.org/wiki/2-2DEngel_group
wiki namespace: 0
wiki page content language: en
defining ingredient: Abelian group
wiki page sort key: 2-Engel group
weaker than: Abelian group

Cyclic group
type: Subject
page: https://groupprops.subwiki.org/wiki/Cyclic_group
wiki namespace: 0
wiki page content language: en
stronger than: Abelian group
wiki page sort key: Cyclic group

Group whose automorphism group is cyclic
type: Subject
page: https://groupprops.subwiki.org/wiki/Group_whose_automorphism_group_is_cyclic
wiki namespace: 0
wiki page content language: en
stronger than: Abelian group
wiki page sort key: Group whose automorphism group is cyclic

Nilpotent group
type: Subject
page: https://groupprops.subwiki.org/wiki/Nilpotent_group
wiki namespace: 0
wiki page content language: en
weaker than: Abelian group
wiki page sort key: Nilpotent group

http://groupprops.subwiki.org/wiki/Special:URIResolver/Cube_map_is_endomorphism_iff_abelian_-28if_order_is_not_a_multiple_of_3-29-23_44c1e186a5819618e75f8b2de8338cf3
type: Subject
master page: Cube map is endomorphism iff abelian (if order is not a multiple of 3)
wiki namespace: 0
wiki page sort key: Cube map is endomorphism iff abelian (if order is not a multiple of 3)#Abelian group# 44c1e186a5819618e75f8b2de8338cf3
page: Abelian group

http://groupprops.subwiki.org/wiki/Special:URIResolver/Cyclic_automorphism_group_implies_abelian-23_44c1e186a5819618e75f8b2de8338cf3
type: Subject
master page: http://groupprops.subwiki.org/wiki/Special:URIResolver/Cyclic_automorphism_group_implies_abelian
wiki namespace: 0
wiki page sort key: Cyclic automorphism group implies abelian#Abelian group# 44c1e186a5819618e75f8b2de8338cf3
page: Abelian group

Dissatisfies property
type: ObjectProperty
page: https://groupprops.subwiki.org/wiki/Property-3ADissatisfies_property
wiki namespace: 132
wiki page content language: en
wiki page sort key: Dissatisfies property
type: wpg

Satisfies property
type: ObjectProperty
page: https://groupprops.subwiki.org/wiki/Property-3ASatisfies_property
wiki namespace: 132
wiki page content language: en
wiki page sort key: Satisfies property
type: wpg

Proves property satisfaction of
type: ObjectProperty
page: https://groupprops.subwiki.org/wiki/Property-3AProves_property_satisfaction_of
wiki namespace: 132
wiki page content language: en
wiki page sort key: Proves property satisfaction of

Analogue of
type: ObjectProperty
page: https://groupprops.subwiki.org/wiki/Property-3AAnalogue_of
wiki namespace: 132
wiki page content language: en
wiki page sort key: Analogue of

Conjunction involving
type: ObjectProperty
page: https://groupprops.subwiki.org/wiki/Property-3AConjunction_involving
wiki namespace: 132
wiki page content language: en
wiki page sort key: Conjunction involving

Defining ingredient
type: ObjectProperty
page: https://groupprops.subwiki.org/wiki/Property-3ADefining_ingredient
wiki namespace: 132
wiki page content language: en
wiki page sort key: Defining ingredient

Stronger than
type: ObjectProperty
page: https://groupprops.subwiki.org/wiki/Property-3AStronger_than
wiki namespace: 132
wiki page content language: en
wiki page sort key: Stronger than

Weaker than
type: ObjectProperty
page: https://groupprops.subwiki.org/wiki/Property-3AWeaker_than
wiki namespace: 132
wiki page content language: en
wiki page sort key: Weaker than

Page
type: ObjectProperty
page: https://groupprops.subwiki.org/wiki/Property-3APage
wiki namespace: 132
wiki page content language: en
wiki page sort key: Page