Category-3AGroup property implications
type: Ontology
creation date: 2026-07-14T00:24:15+00:00
imports: http://semantic-mediawiki.org/swivt/1.0

Group property implications
type: Class
page: https://groupprops.subwiki.org/wiki/Category-3AGroup_property_implications
wiki namespace: 14
wiki page content language: en
wiki page sort key: Group property implications

Dedekind implies ACIC
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Dedekind_implies_ACIC
wiki namespace: 0
wiki page content language: en
wiki page sort key: Dedekind implies ACIC

Prime power order implies nilpotent
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Prime_power_order_implies_nilpotent
wiki namespace: 0
wiki page content language: en
wiki page sort key: Prime power order implies nilpotent

Nilpotent implies normalizer condition
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Nilpotent_implies_normalizer_condition
wiki namespace: 0
wiki page content language: en
wiki page sort key: Nilpotent implies normalizer condition

Cyclic over central implies abelian
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Cyclic_over_central_implies_abelian
wiki namespace: 0
wiki page content language: en
wiki page sort key: Cyclic over central implies abelian

Every finite group has a finite composition series
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Every_finite_group_has_a_finite_composition_series
wiki namespace: 0
wiki page content language: en
wiki page sort key: Every finite group has a finite composition series

P-Frattini-realizable implies not non-abelian cyclic-center
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/P-2DFrattini-2Drealizable_implies_not_non-2Dabelian_cyclic-2Dcenter
wiki namespace: 0
wiki page content language: en
wiki page sort key: P-Frattini-realizable implies not non-abelian cyclic-center

Order has only two prime factors implies solvable
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Order_has_only_two_prime_factors_implies_solvable
wiki namespace: 0
wiki page content language: en
wiki page sort key: Order has only two prime factors implies solvable

Dihedral implies solvable
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Dihedral_implies_solvable
wiki namespace: 0
wiki page content language: en
wiki page sort key: Dihedral implies solvable

Monomial implies solvable
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Monomial_implies_solvable
wiki namespace: 0
wiki page content language: en
wiki page sort key: Monomial implies solvable

Supersolvable implies nilpotent derived subgroup
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Supersolvable_implies_nilpotent_derived_subgroup
wiki namespace: 0
wiki page content language: en
wiki page sort key: Supersolvable implies nilpotent derived subgroup

Abelian implies every element is automorphic to its inverse
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Abelian_implies_every_element_is_automorphic_to_its_inverse
wiki namespace: 0
wiki page content language: en
wiki page sort key: Abelian implies every element is automorphic to its inverse

Finite minimal simple implies 2-generated
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Finite_minimal_simple_implies_2-2Dgenerated
wiki namespace: 0
wiki page content language: en
wiki page sort key: Finite minimal simple implies 2-generated

Odd-order implies solvable
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Odd-2Dorder_implies_solvable
wiki namespace: 0
wiki page content language: en
wiki page sort key: Odd-order implies solvable

Finitely generated and residually finite implies Hopfian
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Finitely_generated_and_residually_finite_implies_Hopfian
wiki namespace: 0
wiki page content language: en
wiki page sort key: Finitely generated and residually finite implies Hopfian

Finitely generated implies finitely many homomorphisms to any finite group
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Finitely_generated_implies_finitely_many_homomorphisms_to_any_finite_group
wiki namespace: 0
wiki page content language: en
wiki page sort key: Finitely generated implies finitely many homomorphisms to any finite group

Linearly primitive implies cyclic-center
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Linearly_primitive_implies_cyclic-2Dcenter
wiki namespace: 0
wiki page content language: en
wiki page sort key: Linearly primitive implies cyclic-center

Ascending chain condition on normal subgroups implies Hopfian
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Ascending_chain_condition_on_normal_subgroups_implies_Hopfian
wiki namespace: 0
wiki page content language: en
wiki page sort key: Ascending chain condition on normal subgroups implies Hopfian

Ascending chain condition on subnormal subgroups implies subnormal join property
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Ascending_chain_condition_on_subnormal_subgroups_implies_subnormal_join_property
wiki namespace: 0
wiki page content language: en
wiki page sort key: Ascending chain condition on subnormal subgroups implies subnormal join property

Frattini-embedded normal-realizable implies every automorph-conjugate subgroup is characteristic
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Frattini-2Dembedded_normal-2Drealizable_implies_every_automorph-2Dconjugate_subgroup_is_characteristic
wiki namespace: 0
wiki page content language: en
wiki page sort key: Frattini-embedded normal-realizable implies every automorph-conjugate subgroup is characteristic

Strongly p-solvable implies Glauberman type for odd p
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Strongly_p-2Dsolvable_implies_Glauberman_type_for_odd_p
wiki namespace: 0
wiki page content language: en
wiki page sort key: Strongly p-solvable implies Glauberman type for odd p

P-solvable implies p-constrained
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/P-2Dsolvable_implies_p-2Dconstrained
wiki namespace: 0
wiki page content language: en
wiki page sort key: P-solvable implies p-constrained

Maximal among abelian normal implies self-centralizing in nilpotent
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Maximal_among_abelian_normal_implies_self-2Dcentralizing_in_nilpotent
wiki namespace: 0
wiki page content language: en
wiki page sort key: Maximal among abelian normal implies self-centralizing in nilpotent

Finitely many homomorphisms to any finite group implies every subgroup of finite index has finitely many automorphic subgroups
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Finitely_many_homomorphisms_to_any_finite_group_implies_every_subgroup_of_finite_index_has_finitely_many_automorphic_subgroups
wiki namespace: 0
wiki page content language: en
wiki page sort key: Finitely many homomorphisms to any finite group implies every subgroup of finite index has finitely many automorphic subgroups

Finite implies subnormal join property
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Finite_implies_subnormal_join_property
wiki namespace: 0
wiki page content language: en
wiki page sort key: Finite implies subnormal join property

Automorphism group is transitive on non-identity elements implies characteristically simple
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Automorphism_group_is_transitive_on_non-2Didentity_elements_implies_characteristically_simple
wiki namespace: 0
wiki page content language: en
wiki page sort key: Automorphism group is transitive on non-identity elements implies characteristically simple

Finite simple implies 2-generated
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Finite_simple_implies_2-2Dgenerated
wiki namespace: 0
wiki page content language: en
wiki page sort key: Finite simple implies 2-generated

Glauberman type implies ZJ-functor controls fusion
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Glauberman_type_implies_ZJ-2Dfunctor_controls_fusion
wiki namespace: 0
wiki page content language: en
wiki page sort key: Glauberman type implies ZJ-functor controls fusion

Finite normal implies amalgam-characteristic
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Finite_normal_implies_amalgam-2Dcharacteristic
wiki namespace: 0
wiki page content language: en
wiki page sort key: Finite normal implies amalgam-characteristic

Conjugacy-separable implies residually finite
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Conjugacy-2Dseparable_implies_residually_finite
wiki namespace: 0
wiki page content language: en
wiki page sort key: Conjugacy-separable implies residually finite

Cyclic quotient of automorphism group by class-preserving automorphism group implies same orbit sizes of conjugacy classes and irreducible representations under automorphism group
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Cyclic_quotient_of_automorphism_group_by_class-2Dpreserving_automorphism_group_implies_same_orbit_sizes_of_conjugacy_classes_and_irreducible_representations_under_a…
wiki namespace: 0
wiki page content language: en
wiki page sort key: Cyclic quotient of automorphism group by class-preserving automorphism group implies same orbit sizes of conjugacy classes and irreducible representations under automorphism group

Nilpotent implies solvable
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Nilpotent_implies_solvable
wiki namespace: 0
wiki page content language: en
wiki page sort key: Nilpotent implies solvable

Nilpotent implies every subgroup is subnormal
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Nilpotent_implies_every_subgroup_is_subnormal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Nilpotent implies every subgroup is subnormal

Every subgroup is subnormal implies normalizer condition
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Every_subgroup_is_subnormal_implies_normalizer_condition
wiki namespace: 0
wiki page content language: en
wiki page sort key: Every subgroup is subnormal implies normalizer condition

Schur multiplier of cyclic group is trivial
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Schur_multiplier_of_cyclic_group_is_trivial
wiki namespace: 0
wiki page content language: en
wiki page sort key: Schur multiplier of cyclic group is trivial

Class-inverting automorphism implies every element is automorphic to its inverse
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Class-2Dinverting_automorphism_implies_every_element_is_automorphic_to_its_inverse
wiki namespace: 0
wiki page content language: en
wiki page sort key: Class-inverting automorphism implies every element is automorphic to its inverse

Cyclic implies abelian
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Cyclic_implies_abelian
wiki namespace: 0
wiki page content language: en
wiki page sort key: Cyclic implies abelian

Nilpotent derived subgroup implies subnormal join property
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Nilpotent_derived_subgroup_implies_subnormal_join_property
wiki namespace: 0
wiki page content language: en
wiki page sort key: Nilpotent derived subgroup implies subnormal join property

Nilpotent implies every nontrivial normal subgroup contains a cyclic normal subgroup
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Nilpotent_implies_every_nontrivial_normal_subgroup_contains_a_cyclic_normal_subgroup
wiki namespace: 0
wiki page content language: en
wiki page sort key: Nilpotent implies every nontrivial normal subgroup contains a cyclic normal subgroup

Nilpotent implies center is normality-large
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Nilpotent_implies_center_is_normality-2Dlarge
wiki namespace: 0
wiki page content language: en
wiki page sort key: Nilpotent implies center is normality-large

Center is normality-large implies every nontrivial normal subgroup contains a cyclic normal subgroup
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Center_is_normality-2Dlarge_implies_every_nontrivial_normal_subgroup_contains_a_cyclic_normal_subgroup
wiki namespace: 0
wiki page content language: en
wiki page sort key: Center is normality-large implies every nontrivial normal subgroup contains a cyclic normal subgroup

Metabelian implies subnormal join property
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Metabelian_implies_subnormal_join_property
wiki namespace: 0
wiki page content language: en
wiki page sort key: Metabelian implies subnormal join property

Finitely generated abelian is subgroup-closed
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Finitely_generated_abelian_is_subgroup-2Dclosed
wiki namespace: 0
wiki page content language: en
wiki page sort key: Finitely generated abelian is subgroup-closed

Frattini-embedded normal-realizable implies inner-in-automorphism-Frattini
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Frattini-2Dembedded_normal-2Drealizable_implies_inner-2Din-2Dautomorphism-2DFrattini
wiki namespace: 0
wiki page content language: en
wiki page sort key: Frattini-embedded normal-realizable implies inner-in-automorphism-Frattini

Extraspecial implies Camina
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Extraspecial_implies_Camina
wiki namespace: 0
wiki page content language: en
wiki page sort key: Extraspecial implies Camina

Prime power order implies subgroups of all orders dividing the group order
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Prime_power_order_implies_subgroups_of_all_orders_dividing_the_group_order
wiki namespace: 0
wiki page content language: en
wiki page sort key: Prime power order implies subgroups of all orders dividing the group order

Schur multiplier of Z-group is trivial
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Schur_multiplier_of_Z-2Dgroup_is_trivial
wiki namespace: 0
wiki page content language: en
wiki page sort key: Schur multiplier of Z-group is trivial

Nilpotent implies every maximal subgroup is normal
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Nilpotent_implies_every_maximal_subgroup_is_normal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Nilpotent implies every maximal subgroup is normal

Normalizer condition implies every maximal subgroup is normal
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Normalizer_condition_implies_every_maximal_subgroup_is_normal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Normalizer condition implies every maximal subgroup is normal

Multiplicative group of a field implies every finite subgroup is cyclic
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Multiplicative_group_of_a_field_implies_every_finite_subgroup_is_cyclic
wiki namespace: 0
wiki page content language: en
wiki page sort key: Multiplicative group of a field implies every finite subgroup is cyclic

At most n elements of order dividing n implies every finite subgroup is cyclic
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/At_most_n_elements_of_order_dividing_n_implies_every_finite_subgroup_is_cyclic
wiki namespace: 0
wiki page content language: en
wiki page sort key: At most n elements of order dividing n implies every finite subgroup is cyclic

Free implies residually nilpotent
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Free_implies_residually_nilpotent
wiki namespace: 0
wiki page content language: en
wiki page sort key: Free implies residually nilpotent

FZ implies generalized subnormal join property
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/FZ_implies_generalized_subnormal_join_property
wiki namespace: 0
wiki page content language: en
wiki page sort key: FZ implies generalized subnormal join property

Abelian implies ACIC
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Abelian_implies_ACIC
wiki namespace: 0
wiki page content language: en
wiki page sort key: Abelian implies ACIC

Locally cyclic implies abelian automorphism group
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Locally_cyclic_implies_abelian_automorphism_group
wiki namespace: 0
wiki page content language: en
wiki page sort key: Locally cyclic implies abelian automorphism group

Artinian implies co-Hopfian
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Artinian_implies_co-2DHopfian
wiki namespace: 0
wiki page content language: en
wiki page sort key: Artinian implies co-Hopfian

Algebra group implies power degree group for field size
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Algebra_group_implies_power_degree_group_for_field_size
wiki namespace: 0
wiki page content language: en
wiki page sort key: Algebra group implies power degree group for field size

Free implies residually finite
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Free_implies_residually_finite
wiki namespace: 0
wiki page content language: en
wiki page sort key: Free implies residually finite

SQ-universal implies no nontrivial identity
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/SQ-2Duniversal_implies_no_nontrivial_identity
wiki namespace: 0
wiki page content language: en
wiki page sort key: SQ-universal implies no nontrivial identity

Additive group of a field implies characteristic in holomorph
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Additive_group_of_a_field_implies_characteristic_in_holomorph
wiki namespace: 0
wiki page content language: en
wiki page sort key: Additive group of a field implies characteristic in holomorph

Class two implies generated by abelian normal subgroups
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Class_two_implies_generated_by_abelian_normal_subgroups
wiki namespace: 0
wiki page content language: en
wiki page sort key: Class two implies generated by abelian normal subgroups

Abelian implies every subgroup is normal
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Abelian_implies_every_subgroup_is_normal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Abelian implies every subgroup is normal

Finite supersolvable implies subgroups of all orders dividing the group order
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Finite_supersolvable_implies_subgroups_of_all_orders_dividing_the_group_order
wiki namespace: 0
wiki page content language: en
wiki page sort key: Finite supersolvable implies subgroups of all orders dividing the group order

Odd-order cyclic group is characteristic in holomorph
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Odd-2Dorder_cyclic_group_is_characteristic_in_holomorph
wiki namespace: 0
wiki page content language: en
wiki page sort key: Odd-order cyclic group is characteristic in holomorph

Finitely generated inner automorphism group implies every locally inner automorphism is inner
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Finitely_generated_inner_automorphism_group_implies_every_locally_inner_automorphism_is_inner
wiki namespace: 0
wiki page content language: en
wiki page sort key: Finitely generated inner automorphism group implies every locally inner automorphism is inner

K implies Frattini-free
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/K_implies_Frattini-2Dfree
wiki namespace: 0
wiki page content language: en
wiki page sort key: K implies Frattini-free

FZ implies finite derived subgroup
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/FZ_implies_finite_derived_subgroup
wiki namespace: 0
wiki page content language: en
wiki page sort key: FZ implies finite derived subgroup

4-Engel implies locally nilpotent for groups
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/4-2DEngel_implies_locally_nilpotent_for_groups
wiki namespace: 0
wiki page content language: en
wiki page sort key: 4-Engel implies locally nilpotent for groups

Normalizer condition implies locally nilpotent
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Normalizer_condition_implies_locally_nilpotent
wiki namespace: 0
wiki page content language: en
wiki page sort key: Normalizer condition implies locally nilpotent

Noetherian implies Hopfian
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Noetherian_implies_Hopfian
wiki namespace: 0
wiki page content language: en
wiki page sort key: Noetherian implies Hopfian

Finitely generated implies countable
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Finitely_generated_implies_countable
wiki namespace: 0
wiki page content language: en
wiki page sort key: Finitely generated implies countable

Nilpotent automorphism group implies nilpotent of class at most one more
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Nilpotent_automorphism_group_implies_nilpotent_of_class_at_most_one_more
wiki namespace: 0
wiki page content language: en
wiki page sort key: Nilpotent automorphism group implies nilpotent of class at most one more

Characteristically simple implies CSCFN-realizable
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Characteristically_simple_implies_CSCFN-2Drealizable
wiki namespace: 0
wiki page content language: en
wiki page sort key: Characteristically simple implies CSCFN-realizable

Solvable automorphism group implies solvable of derived length at most one more
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Solvable_automorphism_group_implies_solvable_of_derived_length_at_most_one_more
wiki namespace: 0
wiki page content language: en
wiki page sort key: Solvable automorphism group implies solvable of derived length at most one more

2-Engel implies class three for groups
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/2-2DEngel_implies_class_three_for_groups
wiki namespace: 0
wiki page content language: en
wiki page sort key: 2-Engel implies class three for groups

Finitely presented implies all homomorphisms to any finite group can be listed in finite time
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Finitely_presented_implies_all_homomorphisms_to_any_finite_group_can_be_listed_in_finite_time
wiki namespace: 0
wiki page content language: en
wiki page sort key: Finitely presented implies all homomorphisms to any finite group can be listed in finite time

Finitely presented and residually finite implies solvable word problem
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Finitely_presented_and_residually_finite_implies_solvable_word_problem
wiki namespace: 0
wiki page content language: en
wiki page sort key: Finitely presented and residually finite implies solvable word problem

Schur multiplier of free group is trivial
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Schur_multiplier_of_free_group_is_trivial
wiki namespace: 0
wiki page content language: en
wiki page sort key: Schur multiplier of free group is trivial

Finitely generated and free implies Hopfian
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Finitely_generated_and_free_implies_Hopfian
wiki namespace: 0
wiki page content language: en
wiki page sort key: Finitely generated and free implies Hopfian

Abelian automorphism group implies class two
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Abelian_automorphism_group_implies_class_two
wiki namespace: 0
wiki page content language: en
wiki page sort key: Abelian automorphism group implies class two

3-Engel implies locally nilpotent for groups
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/3-2DEngel_implies_locally_nilpotent_for_groups
wiki namespace: 0
wiki page content language: en
wiki page sort key: 3-Engel implies locally nilpotent for groups

Primitive implies innately transitive
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Primitive_implies_innately_transitive
wiki namespace: 0
wiki page content language: en
wiki page sort key: Primitive implies innately transitive

Abelian implies self-centralizing in holomorph
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Abelian_implies_self-2Dcentralizing_in_holomorph
wiki namespace: 0
wiki page content language: en
wiki page sort key: Abelian implies self-centralizing in holomorph

3-step group implies solvable CN-group
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/3-2Dstep_group_implies_solvable_CN-2Dgroup
wiki namespace: 0
wiki page content language: en
wiki page sort key: 3-step group implies solvable CN-group

Conjugacy-separable implies every quotient-pullbackable automorphism is class-preserving
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Conjugacy-2Dseparable_implies_every_quotient-2Dpullbackable_automorphism_is_class-2Dpreserving
wiki namespace: 0
wiki page content language: en
wiki page sort key: Conjugacy-separable implies every quotient-pullbackable automorphism is class-preserving

Cyclic implies abelian automorphism group
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Cyclic_implies_abelian_automorphism_group
wiki namespace: 0
wiki page content language: en
wiki page sort key: Cyclic implies abelian automorphism group

Centerless and maximal in automorphism group implies every automorphism is normal-extensible
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Centerless_and_maximal_in_automorphism_group_implies_every_automorphism_is_normal-2Dextensible
wiki namespace: 0
wiki page content language: en
wiki page sort key: Centerless and maximal in automorphism group implies every automorphism is normal-extensible

Conjugacy-separable and aperiodic implies every extensible automorphism is class-preserving
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Conjugacy-2Dseparable_and_aperiodic_implies_every_extensible_automorphism_is_class-2Dpreserving
wiki namespace: 0
wiki page content language: en
wiki page sort key: Conjugacy-separable and aperiodic implies every extensible automorphism is class-preserving

Finitely presented and conjugacy-separable implies solvable conjugacy problem
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Finitely_presented_and_conjugacy-2Dseparable_implies_solvable_conjugacy_problem
wiki namespace: 0
wiki page content language: en
wiki page sort key: Finitely presented and conjugacy-separable implies solvable conjugacy problem

Proper and normal in quasisimple implies central
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Proper_and_normal_in_quasisimple_implies_central
wiki namespace: 0
wiki page content language: en
wiki page sort key: Proper and normal in quasisimple implies central

Abelian implies nilpotent
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Abelian_implies_nilpotent
wiki namespace: 0
wiki page content language: en
wiki page sort key: Abelian implies nilpotent

Dedekind implies class two
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Dedekind_implies_class_two
wiki namespace: 0
wiki page content language: en
wiki page sort key: Dedekind implies class two

Finite abelian implies same orbit sizes of conjugacy classes and irreducible representations under automorphism group
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Finite_abelian_implies_same_orbit_sizes_of_conjugacy_classes_and_irreducible_representations_under_automorphism_group
wiki namespace: 0
wiki page content language: en
wiki page sort key: Finite abelian implies same orbit sizes of conjugacy classes and irreducible representations under automorphism group

Free implies every subgroup is descendant
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Free_implies_every_subgroup_is_descendant
wiki namespace: 0
wiki page content language: en
wiki page sort key: Free implies every subgroup is descendant

ACIC implies nilpotent (finite groups)
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/ACIC_implies_nilpotent_-28finite_groups-29
wiki namespace: 0
wiki page content language: en
wiki page sort key: ACIC implies nilpotent (finite groups)

Finitely generated abelian implies Hopfian
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Finitely_generated_abelian_implies_Hopfian
wiki namespace: 0
wiki page content language: en
wiki page sort key: Finitely generated abelian implies Hopfian

Normal implies modular
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Normal_implies_modular
wiki namespace: 0
wiki page content language: en
wiki page sort key: Normal implies modular

Locally finite Artinian p-group implies hypercentral
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Locally_finite_Artinian_p-2Dgroup_implies_hypercentral
wiki namespace: 0
wiki page content language: en
wiki page sort key: Locally finite Artinian p-group implies hypercentral

Identity functor controls strong fusion for saturated fusion system on abelian group
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Identity_functor_controls_strong_fusion_for_saturated_fusion_system_on_abelian_group
wiki namespace: 0
wiki page content language: en
wiki page sort key: Identity functor controls strong fusion for saturated fusion system on abelian group

Algebraically closed implies simple
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Algebraically_closed_implies_simple
wiki namespace: 0
wiki page content language: en
wiki page sort key: Algebraically closed implies simple

Finitely generated and nilpotent implies Hopfian
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Finitely_generated_and_nilpotent_implies_Hopfian
wiki namespace: 0
wiki page content language: en
wiki page sort key: Finitely generated and nilpotent implies Hopfian

Finitely generated abelian implies residually finite
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Finitely_generated_abelian_implies_residually_finite
wiki namespace: 0
wiki page content language: en
wiki page sort key: Finitely generated abelian implies residually finite

Noetherian implies subnormal join property
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Noetherian_implies_subnormal_join_property
wiki namespace: 0
wiki page content language: en
wiki page sort key: Noetherian implies subnormal join property

Supersolvable implies every nontrivial normal subgroup contains a cyclic normal subgroup
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Supersolvable_implies_every_nontrivial_normal_subgroup_contains_a_cyclic_normal_subgroup
wiki namespace: 0
wiki page content language: en
wiki page sort key: Supersolvable implies every nontrivial normal subgroup contains a cyclic normal subgroup

Finitely generated implies every subgroup of finite index has finitely many automorphic subgroups
type: Subject
type: Group property implications
page: https://groupprops.subwiki.org/wiki/Finitely_generated_implies_every_subgroup_of_finite_index_has_finitely_many_automorphic_subgroups
wiki namespace: 0
wiki page content language: en
wiki page sort key: Finitely generated implies every subgroup of finite index has finitely many automorphic subgroups

INST
type: DatatypeProperty
page: https://groupprops.subwiki.org/wiki/Property-3A_INST
wiki namespace: 132
wiki page content language: en
wiki page sort key: INST