Category-3ASubgroup property implications
type: Ontology
creation date: 2026-06-06T09:49:13+00:00
imports: http://semantic-mediawiki.org/swivt/1.0

Subgroup property implications
type: Class
page: https://groupprops.subwiki.org/wiki/Category-3ASubgroup_property_implications
wiki namespace: 14
wiki page content language: en
wiki page modification date: 2008-05-07T22:39:22Z
modification date#aux: 2454594.4440046
wiki page sort key: Subgroup property implications
subclass of: Category:Property implications

WNSCDIN implies every normalizer-relatively normal conjugation-invariantly relatively normal subgroup is weakly closed
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/WNSCDIN_implies_every_normalizer-2Drelatively_normal_conjugation-2Dinvariantly_relatively_normal_subgroup_is_weakly_closed
wiki namespace: 0
wiki page content language: en
wiki page sort key: WNSCDIN implies every normalizer-relatively normal conjugation-invariantly relatively normal subgroup is weakly closed

Characteristic implies normal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Characteristic_implies_normal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Characteristic implies normal

Sylow normalizer implies weakly abnormal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Sylow_normalizer_implies_weakly_abnormal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Sylow normalizer implies weakly abnormal

Subgroup of index two is normal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Subgroup_of_index_two_is_normal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Subgroup of index two is normal

Central factor implies transitively normal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Central_factor_implies_transitively_normal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Central factor implies transitively normal

Normal Hall implies permutably complemented
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Normal_Hall_implies_permutably_complemented
wiki namespace: 0
wiki page content language: en
wiki page sort key: Normal Hall implies permutably complemented

Intermediately automorph-conjugate of normal implies weakly pronormal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Intermediately_automorph-2Dconjugate_of_normal_implies_weakly_pronormal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Intermediately automorph-conjugate of normal implies weakly pronormal

Characteristic upper-hook AEP implies characteristic
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Characteristic_upper-2Dhook_AEP_implies_characteristic
wiki namespace: 0
wiki page content language: en
wiki page sort key: Characteristic upper-hook AEP implies characteristic

Normal and self-centralizing implies coprime automorphism-faithful
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Normal_and_self-2Dcentralizing_implies_coprime_automorphism-2Dfaithful
wiki namespace: 0
wiki page content language: en
wiki page sort key: Normal and self-centralizing implies coprime automorphism-faithful

Prime power order implies center is normality-large
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Prime_power_order_implies_center_is_normality-2Dlarge
wiki namespace: 0
wiki page content language: en
wiki page sort key: Prime power order implies center is normality-large

Artinian implies periodic
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Artinian_implies_periodic
wiki namespace: 0
wiki page content language: en
wiki page sort key: Artinian implies periodic

Conjugate-join-closed subnormal implies join-transitively subnormal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Conjugate-2Djoin-2Dclosed_subnormal_implies_join-2Dtransitively_subnormal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Conjugate-join-closed subnormal implies join-transitively subnormal

Amalgam-characteristic implies potentially characteristic
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Amalgam-2Dcharacteristic_implies_potentially_characteristic
wiki namespace: 0
wiki page content language: en
wiki page sort key: Amalgam-characteristic implies potentially characteristic

Central implies normal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Central_implies_normal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Central implies normal

Permutable implies locally subnormal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Permutable_implies_locally_subnormal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Permutable implies locally subnormal

Fully invariant implies characteristic
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Fully_invariant_implies_characteristic
wiki namespace: 0
wiki page content language: en
wiki page sort key: Fully invariant implies characteristic

Pronormal implies WNSCDIN
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Pronormal_implies_WNSCDIN
wiki namespace: 0
wiki page content language: en
wiki page sort key: Pronormal implies WNSCDIN

Maximal among abelian normal implies self-centralizing in nilpotent
type: Subject
type: Subgroup 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

Double coset index two implies maximal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Double_coset_index_two_implies_maximal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Double coset index two implies maximal

Subgroup of finite index has a left transversal that is also a right transversal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Subgroup_of_finite_index_has_a_left_transversal_that_is_also_a_right_transversal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Subgroup of finite index has a left transversal that is also a right transversal

Subgroup of finite group has a left transversal that is also a right transversal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Subgroup_of_finite_group_has_a_left_transversal_that_is_also_a_right_transversal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Subgroup of finite group has a left transversal that is also a right transversal

Central factor implies normal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Central_factor_implies_normal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Central factor implies normal

Commutator of a transitively normal subgroup and a subset implies normal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Commutator_of_a_transitively_normal_subgroup_and_a_subset_implies_normal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Commutator of a transitively normal subgroup and a subset implies normal

Left-transitively complemented normal implies characteristic
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Left-2Dtransitively_complemented_normal_implies_characteristic
wiki namespace: 0
wiki page content language: en
wiki page sort key: Left-transitively complemented normal implies characteristic

Pronormal implies MWNSCDIN
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Pronormal_implies_MWNSCDIN
wiki namespace: 0
wiki page content language: en
wiki page sort key: Pronormal implies MWNSCDIN

Base of a wreath product implies right-transitively conjugate-permutable
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Base_of_a_wreath_product_implies_right-2Dtransitively_conjugate-2Dpermutable
wiki namespace: 0
wiki page content language: en
wiki page sort key: Base of a wreath product implies right-transitively conjugate-permutable

Pronormal implies self-conjugate-permutable
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Pronormal_implies_self-2Dconjugate-2Dpermutable
wiki namespace: 0
wiki page content language: en
wiki page sort key: Pronormal implies self-conjugate-permutable

Monolith is strictly characteristic
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Monolith_is_strictly_characteristic
wiki namespace: 0
wiki page content language: en
wiki page sort key: Monolith is strictly characteristic

Direct factor implies normal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Direct_factor_implies_normal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Direct factor implies normal

Central implies normal satisfying the subgroup-to-quotient powering-invariance implication
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Central_implies_normal_satisfying_the_subgroup-2Dto-2Dquotient_powering-2Dinvariance_implication
wiki namespace: 0
wiki page content language: en
wiki page sort key: Central implies normal satisfying the subgroup-to-quotient powering-invariance implication

Central implies image-potentially characteristic
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Central_implies_image-2Dpotentially_characteristic
wiki namespace: 0
wiki page content language: en
wiki page sort key: Central implies image-potentially characteristic

Central implies amalgam-characteristic
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Central_implies_amalgam-2Dcharacteristic
wiki namespace: 0
wiki page content language: en
wiki page sort key: Central implies amalgam-characteristic

Amalgam-characteristic implies image-potentially characteristic
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Amalgam-2Dcharacteristic_implies_image-2Dpotentially_characteristic
wiki namespace: 0
wiki page content language: en
wiki page sort key: Amalgam-characteristic implies image-potentially characteristic

Finite normal implies amalgam-characteristic
type: Subject
type: Subgroup 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

Verbal implies fully invariant
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Verbal_implies_fully_invariant
wiki namespace: 0
wiki page content language: en
wiki page sort key: Verbal implies fully invariant

Direct factor implies join-transitively central factor
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Direct_factor_implies_join-2Dtransitively_central_factor
wiki namespace: 0
wiki page content language: en
wiki page sort key: Direct factor implies join-transitively central factor

Direct factor implies right-quotient-transitively central factor
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Direct_factor_implies_right-2Dquotient-2Dtransitively_central_factor
wiki namespace: 0
wiki page content language: en
wiki page sort key: Direct factor implies right-quotient-transitively central factor

Right-quotient-transitively central factor implies join-transitively central factor
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Right-2Dquotient-2Dtransitively_central_factor_implies_join-2Dtransitively_central_factor
wiki namespace: 0
wiki page content language: en
wiki page sort key: Right-quotient-transitively central factor implies join-transitively central factor

Normal implies permutable
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Normal_implies_permutable
wiki namespace: 0
wiki page content language: en
wiki page sort key: Normal implies permutable

Iterated agemo implies verbal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Iterated_agemo_implies_verbal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Iterated agemo implies verbal

Endomorphism kernel implies quotient-powering-invariant
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Endomorphism_kernel_implies_quotient-2Dpowering-2Dinvariant
wiki namespace: 0
wiki page content language: en
wiki page sort key: Endomorphism kernel implies quotient-powering-invariant

Endomorphism image implies powering-invariant
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Endomorphism_image_implies_powering-2Dinvariant
wiki namespace: 0
wiki page content language: en
wiki page sort key: Endomorphism image implies powering-invariant

Endomorphism image implies divisibility-closed
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Endomorphism_image_implies_divisibility-2Dclosed
wiki namespace: 0
wiki page content language: en
wiki page sort key: Endomorphism image implies divisibility-closed

Divisibility-closed implies powering-invariant
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Divisibility-2Dclosed_implies_powering-2Dinvariant
wiki namespace: 0
wiki page content language: en
wiki page sort key: Divisibility-closed implies powering-invariant

Finite normal implies image-potentially characteristic
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Finite_normal_implies_image-2Dpotentially_characteristic
wiki namespace: 0
wiki page content language: en
wiki page sort key: Finite normal implies image-potentially characteristic

Hall retract implies order-conjugate
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Hall_retract_implies_order-2Dconjugate
wiki namespace: 0
wiki page content language: en
wiki page sort key: Hall retract implies order-conjugate

Central implies potentially characteristic
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Central_implies_potentially_characteristic
wiki namespace: 0
wiki page content language: en
wiki page sort key: Central implies potentially characteristic

Normal and centralizer-free implies automorphism-faithful
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Normal_and_centralizer-2Dfree_implies_automorphism-2Dfaithful
wiki namespace: 0
wiki page content language: en
wiki page sort key: Normal and centralizer-free implies automorphism-faithful

Cocentral implies central factor
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Cocentral_implies_central_factor
wiki namespace: 0
wiki page content language: en
wiki page sort key: Cocentral implies central factor

3-subnormal implies finite-conjugate-join-closed subnormal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/3-2Dsubnormal_implies_finite-2Dconjugate-2Djoin-2Dclosed_subnormal
wiki namespace: 0
wiki page content language: en
wiki page sort key: 3-subnormal implies finite-conjugate-join-closed subnormal

2-subnormal implies join-transitively subnormal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/2-2Dsubnormal_implies_join-2Dtransitively_subnormal
wiki namespace: 0
wiki page content language: en
wiki page sort key: 2-subnormal implies join-transitively subnormal

2-subnormal implies conjugate-permutable
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/2-2Dsubnormal_implies_conjugate-2Dpermutable
wiki namespace: 0
wiki page content language: en
wiki page sort key: 2-subnormal implies conjugate-permutable

Perfect subnormal implies join-transitively subnormal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Perfect_subnormal_implies_join-2Dtransitively_subnormal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Perfect subnormal implies join-transitively subnormal

Perfect subnormal implies subnormal-permutable
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Perfect_subnormal_implies_subnormal-2Dpermutable
wiki namespace: 0
wiki page content language: en
wiki page sort key: Perfect subnormal implies subnormal-permutable

Self-centralizing and minimal normal implies strictly characteristic
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Self-2Dcentralizing_and_minimal_normal_implies_strictly_characteristic
wiki namespace: 0
wiki page content language: en
wiki page sort key: Self-centralizing and minimal normal implies strictly characteristic

Direct factor implies central factor
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Direct_factor_implies_central_factor
wiki namespace: 0
wiki page content language: en
wiki page sort key: Direct factor implies central factor

Join of finitely many direct factors implies central factor
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Join_of_finitely_many_direct_factors_implies_central_factor
wiki namespace: 0
wiki page content language: en
wiki page sort key: Join of finitely many direct factors implies central factor

Characteristic central factor of WNSCDIN implies WNSCDIN
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Characteristic_central_factor_of_WNSCDIN_implies_WNSCDIN
wiki namespace: 0
wiki page content language: en
wiki page sort key: Characteristic central factor of WNSCDIN implies WNSCDIN

Paranormal implies weakly normal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Paranormal_implies_weakly_normal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Paranormal implies weakly normal

Paranormal implies polynormal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Paranormal_implies_polynormal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Paranormal implies polynormal

Permutable implies ascendant
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Permutable_implies_ascendant
wiki namespace: 0
wiki page content language: en
wiki page sort key: Permutable implies ascendant

Pronormal and subnormal implies normal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Pronormal_and_subnormal_implies_normal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Pronormal and subnormal implies normal

Pronormal implies intermediately subnormal-to-normal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Pronormal_implies_intermediately_subnormal-2Dto-2Dnormal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Pronormal implies intermediately subnormal-to-normal

Subgroup-cofactorial automorphism-invariant implies left-transitively 2-subnormal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Subgroup-2Dcofactorial_automorphism-2Dinvariant_implies_left-2Dtransitively_2-2Dsubnormal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Subgroup-cofactorial automorphism-invariant implies left-transitively 2-subnormal

Complemented normal implies endomorphism kernel
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Complemented_normal_implies_endomorphism_kernel
wiki namespace: 0
wiki page content language: en
wiki page sort key: Complemented normal implies endomorphism kernel

Maximal implies pronormal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Maximal_implies_pronormal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Maximal implies pronormal

Join with any distinct conjugate is the whole group implies pronormal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Join_with_any_distinct_conjugate_is_the_whole_group_implies_pronormal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Join with any distinct conjugate is the whole group implies pronormal

Central implies potentially verbal in finite
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Central_implies_potentially_verbal_in_finite
wiki namespace: 0
wiki page content language: en
wiki page sort key: Central implies potentially verbal in finite

Periodic normal implies potentially characteristic
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Periodic_normal_implies_potentially_characteristic
wiki namespace: 0
wiki page content language: en
wiki page sort key: Periodic normal implies potentially characteristic

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

Retract implies local divisibility-invariant
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Retract_implies_local_divisibility-2Dinvariant
wiki namespace: 0
wiki page content language: en
wiki page sort key: Retract implies local divisibility-invariant

Nilpotent implies every maximal subgroup is normal
type: Subject
type: Subgroup 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

Fully invariant direct factor implies left-transitively homomorph-containing
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Fully_invariant_direct_factor_implies_left-2Dtransitively_homomorph-2Dcontaining
wiki namespace: 0
wiki page content language: en
wiki page sort key: Fully invariant direct factor implies left-transitively homomorph-containing

Central subgroup implies join-transitively central factor
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Central_subgroup_implies_join-2Dtransitively_central_factor
wiki namespace: 0
wiki page content language: en
wiki page sort key: Central subgroup implies join-transitively central factor

Cyclic-quotient characteristic implies upward-closed characteristic
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Cyclic-2Dquotient_characteristic_implies_upward-2Dclosed_characteristic
wiki namespace: 0
wiki page content language: en
wiki page sort key: Cyclic-quotient characteristic implies upward-closed characteristic

Cyclic characteristic implies hereditarily characteristic
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Cyclic_characteristic_implies_hereditarily_characteristic
wiki namespace: 0
wiki page content language: en
wiki page sort key: Cyclic characteristic implies hereditarily characteristic

Cocentral implies centralizer-dense
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Cocentral_implies_centralizer-2Ddense
wiki namespace: 0
wiki page content language: en
wiki page sort key: Cocentral implies centralizer-dense

Maximal implies modular
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Maximal_implies_modular
wiki namespace: 0
wiki page content language: en
wiki page sort key: Maximal implies modular

Normal and self-centralizing implies normality-large
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Normal_and_self-2Dcentralizing_implies_normality-2Dlarge
wiki namespace: 0
wiki page content language: en
wiki page sort key: Normal and self-centralizing implies normality-large

NE implies weakly normal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/NE_implies_weakly_normal
wiki namespace: 0
wiki page content language: en
wiki page sort key: NE implies weakly normal

Isomorph-containing implies characteristic
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Isomorph-2Dcontaining_implies_characteristic
wiki namespace: 0
wiki page content language: en
wiki page sort key: Isomorph-containing implies characteristic

Strongly paranormal implies paranormal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Strongly_paranormal_implies_paranormal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Strongly paranormal implies paranormal

Commutator of a normal subgroup and a subset implies 2-subnormal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Commutator_of_a_normal_subgroup_and_a_subset_implies_2-2Dsubnormal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Commutator of a normal subgroup and a subset implies 2-subnormal

Normalizer of intermediately subnormal-to-normal implies self-normalizing
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Normalizer_of_intermediately_subnormal-2Dto-2Dnormal_implies_self-2Dnormalizing
wiki namespace: 0
wiki page content language: en
wiki page sort key: Normalizer of intermediately subnormal-to-normal implies self-normalizing

Left-transitively permutable implies characteristic
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Left-2Dtransitively_permutable_implies_characteristic
wiki namespace: 0
wiki page content language: en
wiki page sort key: Left-transitively permutable implies characteristic

Characteristic of CDIN implies CDIN
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Characteristic_of_CDIN_implies_CDIN
wiki namespace: 0
wiki page content language: en
wiki page sort key: Characteristic of CDIN implies CDIN

Normal of finite index implies quotient-powering-invariant
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Normal_of_finite_index_implies_quotient-2Dpowering-2Dinvariant
wiki namespace: 0
wiki page content language: en
wiki page sort key: Normal of finite index implies quotient-powering-invariant

Cyclic normal implies hereditarily normal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Cyclic_normal_implies_hereditarily_normal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Cyclic normal implies hereditarily normal

Fully invariant of strictly characteristic implies strictly characteristic
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Fully_invariant_of_strictly_characteristic_implies_strictly_characteristic
wiki namespace: 0
wiki page content language: en
wiki page sort key: Fully invariant of strictly characteristic implies strictly characteristic

Intersection of DPICF and direct factor is central factor
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Intersection_of_DPICF_and_direct_factor_is_central_factor
wiki namespace: 0
wiki page content language: en
wiki page sort key: Intersection of DPICF and direct factor is central factor

Monolith is characteristic
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Monolith_is_characteristic
wiki namespace: 0
wiki page content language: en
wiki page sort key: Monolith is characteristic

Retract implies central factor of normalizer
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Retract_implies_central_factor_of_normalizer
wiki namespace: 0
wiki page content language: en
wiki page sort key: Retract implies central factor of normalizer

Subgroup of finite abelian group implies finite-abelian-pi-potentially verbal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Subgroup_of_finite_abelian_group_implies_finite-2Dabelian-2Dpi-2Dpotentially_verbal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Subgroup of finite abelian group implies finite-abelian-pi-potentially verbal

Hall-extensible implies class-preserving
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Hall-2Dextensible_implies_class-2Dpreserving
wiki namespace: 0
wiki page content language: en
wiki page sort key: Hall-extensible implies class-preserving

Center of pronormal implies SCDIN
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Center_of_pronormal_implies_SCDIN
wiki namespace: 0
wiki page content language: en
wiki page sort key: Center of pronormal implies SCDIN

Abelian and pronormal implies SCDIN
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Abelian_and_pronormal_implies_SCDIN
wiki namespace: 0
wiki page content language: en
wiki page sort key: Abelian and pronormal implies SCDIN

Finite normal implies quotient-powering-invariant
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Finite_normal_implies_quotient-2Dpowering-2Dinvariant
wiki namespace: 0
wiki page content language: en
wiki page sort key: Finite normal implies quotient-powering-invariant

Commutator of a group and a subgroup implies normal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Commutator_of_a_group_and_a_subgroup_implies_normal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Commutator of a group and a subgroup implies normal

Potentially characteristic implies normal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Potentially_characteristic_implies_normal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Potentially characteristic implies normal

Prehomomorph-contained implies strictly characteristic
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Prehomomorph-2Dcontained_implies_strictly_characteristic
wiki namespace: 0
wiki page content language: en
wiki page sort key: Prehomomorph-contained implies strictly characteristic

Polynormal implies intermediately subnormal-to-normal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Polynormal_implies_intermediately_subnormal-2Dto-2Dnormal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Polynormal implies intermediately subnormal-to-normal

Identity functor controls strong fusion for abelian Sylow subgroup
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Identity_functor_controls_strong_fusion_for_abelian_Sylow_subgroup
wiki namespace: 0
wiki page content language: en
wiki page sort key: Identity functor controls strong fusion for abelian Sylow subgroup

Sylow implies MWNSCDIN
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Sylow_implies_MWNSCDIN
wiki namespace: 0
wiki page content language: en
wiki page sort key: Sylow implies MWNSCDIN

Hall implies paracharacteristic
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Hall_implies_paracharacteristic
wiki namespace: 0
wiki page content language: en
wiki page sort key: Hall implies paracharacteristic

Hall implies join of Sylow subgroups
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Hall_implies_join_of_Sylow_subgroups
wiki namespace: 0
wiki page content language: en
wiki page sort key: Hall implies join of Sylow subgroups

Characteristic subgroup of abelian group implies powering-invariant
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Characteristic_subgroup_of_abelian_group_implies_powering-2Dinvariant
wiki namespace: 0
wiki page content language: en
wiki page sort key: Characteristic subgroup of abelian group implies powering-invariant

Finite implies powering-invariant
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Finite_implies_powering-2Dinvariant
wiki namespace: 0
wiki page content language: en
wiki page sort key: Finite implies powering-invariant

Permutable implies modular
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Permutable_implies_modular
wiki namespace: 0
wiki page content language: en
wiki page sort key: Permutable implies modular

Focal subgroup theorem
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Focal_subgroup_theorem
wiki namespace: 0
wiki page content language: en
wiki page sort key: Focal subgroup theorem

Base of a wreath product implies right-transitively 2-subnormal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Base_of_a_wreath_product_implies_right-2Dtransitively_2-2Dsubnormal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Base of a wreath product implies right-transitively 2-subnormal

Retract implies derived subgroup equals intersection with whole derived subgroup
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Retract_implies_derived_subgroup_equals_intersection_with_whole_derived_subgroup
wiki namespace: 0
wiki page content language: en
wiki page sort key: Retract implies derived subgroup equals intersection with whole derived subgroup

Kernel of a characteristic action on an abelian group implies strongly image-potentially characteristic
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Kernel_of_a_characteristic_action_on_an_abelian_group_implies_strongly_image-2Dpotentially_characteristic
wiki namespace: 0
wiki page content language: en
wiki page sort key: Kernel of a characteristic action on an abelian group implies strongly image-potentially characteristic

Index four implies 2-subnormal or double coset index two
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Index_four_implies_2-2Dsubnormal_or_double_coset_index_two
wiki namespace: 0
wiki page content language: en
wiki page sort key: Index four implies 2-subnormal or double coset index two

Pronormal implies weakly normal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Pronormal_implies_weakly_normal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Pronormal implies weakly normal

Marginal implies direct power-closed characteristic
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Marginal_implies_direct_power-2Dclosed_characteristic
wiki namespace: 0
wiki page content language: en
wiki page sort key: Marginal implies direct power-closed characteristic

Left-transitively 2-subnormal implies normal of characteristic
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Left-2Dtransitively_2-2Dsubnormal_implies_normal_of_characteristic
wiki namespace: 0
wiki page content language: en
wiki page sort key: Left-transitively 2-subnormal implies normal of characteristic

Normal-homomorph-containing implies strictly characteristic
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Normal-2Dhomomorph-2Dcontaining_implies_strictly_characteristic
wiki namespace: 0
wiki page content language: en
wiki page sort key: Normal-homomorph-containing implies strictly characteristic

Periodic normal implies image-potentially characteristic
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Periodic_normal_implies_image-2Dpotentially_characteristic
wiki namespace: 0
wiki page content language: en
wiki page sort key: Periodic normal implies image-potentially characteristic

Sylow implies order-conjugate
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Sylow_implies_order-2Dconjugate
wiki namespace: 0
wiki page content language: en
wiki page sort key: Sylow implies order-conjugate

Normality-preserving endomorphism-invariant implies finite direct power-closed characteristic
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Normality-2Dpreserving_endomorphism-2Dinvariant_implies_finite_direct_power-2Dclosed_characteristic
wiki namespace: 0
wiki page content language: en
wiki page sort key: Normality-preserving endomorphism-invariant implies finite direct power-closed characteristic

Normality-preserving endomorphism-invariant implies characteristic
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Normality-2Dpreserving_endomorphism-2Dinvariant_implies_characteristic
wiki namespace: 0
wiki page content language: en
wiki page sort key: Normality-preserving endomorphism-invariant implies characteristic

Sylow and TI implies CDIN
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Sylow_and_TI_implies_CDIN
wiki namespace: 0
wiki page content language: en
wiki page sort key: Sylow and TI implies CDIN

CDIN of conjugacy-closed implies CDIN
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/CDIN_of_conjugacy-2Dclosed_implies_CDIN
wiki namespace: 0
wiki page content language: en
wiki page sort key: CDIN of conjugacy-closed implies CDIN

Base of a wreath product implies subset-conjugacy-closed
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Base_of_a_wreath_product_implies_subset-2Dconjugacy-2Dclosed
wiki namespace: 0
wiki page content language: en
wiki page sort key: Base of a wreath product implies subset-conjugacy-closed

SCDIN of subset-conjugacy-closed implies SCDIN
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/SCDIN_of_subset-2Dconjugacy-2Dclosed_implies_SCDIN
wiki namespace: 0
wiki page content language: en
wiki page sort key: SCDIN of subset-conjugacy-closed implies SCDIN

Finite index implies completely divisibility-closed
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Finite_index_implies_completely_divisibility-2Dclosed
wiki namespace: 0
wiki page content language: en
wiki page sort key: Finite index implies completely divisibility-closed

Nilpotent-quotient implies subgroup-to-quotient powering-invariance implication
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Nilpotent-2Dquotient_implies_subgroup-2Dto-2Dquotient_powering-2Dinvariance_implication
wiki namespace: 0
wiki page content language: en
wiki page sort key: Nilpotent-quotient implies subgroup-to-quotient powering-invariance implication

Self-centralizing and minimal normal implies characteristic
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Self-2Dcentralizing_and_minimal_normal_implies_characteristic
wiki namespace: 0
wiki page content language: en
wiki page sort key: Self-centralizing and minimal normal implies characteristic

Finite implies local powering-invariant
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Finite_implies_local_powering-2Dinvariant
wiki namespace: 0
wiki page content language: en
wiki page sort key: Finite implies local powering-invariant

Normal subgroup of nilpotent group satisfies the subgroup-to-quotient powering-invariance implication
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Normal_subgroup_of_nilpotent_group_satisfies_the_subgroup-2Dto-2Dquotient_powering-2Dinvariance_implication
wiki namespace: 0
wiki page content language: en
wiki page sort key: Normal subgroup of nilpotent group satisfies the subgroup-to-quotient powering-invariance implication

Normal subgroup contained in the hypercenter satisfies the subgroup-to-quotient powering-invariance implication
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Normal_subgroup_contained_in_the_hypercenter_satisfies_the_subgroup-2Dto-2Dquotient_powering-2Dinvariance_implication
wiki namespace: 0
wiki page content language: en
wiki page sort key: Normal subgroup contained in the hypercenter satisfies the subgroup-to-quotient powering-invariance implication

Intersection of kernels of bihomomorphisms implies completely divisibility-closed
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Intersection_of_kernels_of_bihomomorphisms_implies_completely_divisibility-2Dclosed
wiki namespace: 0
wiki page content language: en
wiki page sort key: Intersection of kernels of bihomomorphisms implies completely divisibility-closed

Kernel of a bihomomorphism implies completely divisibility-closed
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Kernel_of_a_bihomomorphism_implies_completely_divisibility-2Dclosed
wiki namespace: 0
wiki page content language: en
wiki page sort key: Kernel of a bihomomorphism implies completely divisibility-closed

Characteristic and self-centralizing implies coprime automorphism-faithful
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Characteristic_and_self-2Dcentralizing_implies_coprime_automorphism-2Dfaithful
wiki namespace: 0
wiki page content language: en
wiki page sort key: Characteristic and self-centralizing implies coprime automorphism-faithful

Characteristic subgroup of abelian group implies intermediately powering-invariant
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Characteristic_subgroup_of_abelian_group_implies_intermediately_powering-2Dinvariant
wiki namespace: 0
wiki page content language: en
wiki page sort key: Characteristic subgroup of abelian group implies intermediately powering-invariant

2-subnormal implies conjugate-join-closed subnormal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/2-2Dsubnormal_implies_conjugate-2Djoin-2Dclosed_subnormal
wiki namespace: 0
wiki page content language: en
wiki page sort key: 2-subnormal implies conjugate-join-closed subnormal

Index three implies normal or double coset index two
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Index_three_implies_normal_or_double_coset_index_two
wiki namespace: 0
wiki page content language: en
wiki page sort key: Index three implies normal or double coset index two

Subhomomorph-containing implies right-transitively homomorph-containing
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Subhomomorph-2Dcontaining_implies_right-2Dtransitively_homomorph-2Dcontaining
wiki namespace: 0
wiki page content language: en
wiki page sort key: Subhomomorph-containing implies right-transitively homomorph-containing

Central implies abelian normal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Central_implies_abelian_normal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Central implies abelian normal

Conjugacy-closed implies focal subgroup equals derived subgroup
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Conjugacy-2Dclosed_implies_focal_subgroup_equals_derived_subgroup
wiki namespace: 0
wiki page content language: en
wiki page sort key: Conjugacy-closed implies focal subgroup equals derived subgroup

Sylow implies automorph-conjugate
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Sylow_implies_automorph-2Dconjugate
wiki namespace: 0
wiki page content language: en
wiki page sort key: Sylow implies automorph-conjugate

Intermediately normal-to-characteristic implies intermediately subnormal-to-normal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Intermediately_normal-2Dto-2Dcharacteristic_implies_intermediately_subnormal-2Dto-2Dnormal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Intermediately normal-to-characteristic implies intermediately subnormal-to-normal

Finite solvable-extensible implies class-preserving
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Finite_solvable-2Dextensible_implies_class-2Dpreserving
wiki namespace: 0
wiki page content language: en
wiki page sort key: Finite solvable-extensible implies class-preserving

Normal implies join-transitively subnormal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Normal_implies_join-2Dtransitively_subnormal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Normal implies join-transitively subnormal

Permutable and subnormal implies join-transitively subnormal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Permutable_and_subnormal_implies_join-2Dtransitively_subnormal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Permutable and subnormal implies join-transitively subnormal

Normal-potentially characteristic implies normal-extensible automorphism-invariant
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Normal-2Dpotentially_characteristic_implies_normal-2Dextensible_automorphism-2Dinvariant
wiki namespace: 0
wiki page content language: en
wiki page sort key: Normal-potentially characteristic implies normal-extensible automorphism-invariant

Free factor implies self-normalizing or trivial
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Free_factor_implies_self-2Dnormalizing_or_trivial
wiki namespace: 0
wiki page content language: en
wiki page sort key: Free factor implies self-normalizing or trivial

Sylow and TI implies SCDIN
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Sylow_and_TI_implies_SCDIN
wiki namespace: 0
wiki page content language: en
wiki page sort key: Sylow and TI implies SCDIN

Abnormal implies WNSCC
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Abnormal_implies_WNSCC
wiki namespace: 0
wiki page content language: en
wiki page sort key: Abnormal implies WNSCC

Characteristic implies automorph-conjugate
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Characteristic_implies_automorph-2Dconjugate
wiki namespace: 0
wiki page content language: en
wiki page sort key: Characteristic implies automorph-conjugate

Kernel of a bihomomorphism implies abelian-quotient
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Kernel_of_a_bihomomorphism_implies_abelian-2Dquotient
wiki namespace: 0
wiki page content language: en
wiki page sort key: Kernel of a bihomomorphism implies abelian-quotient

Cocentral implies right-quotient-transitively central factor
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Cocentral_implies_right-2Dquotient-2Dtransitively_central_factor
wiki namespace: 0
wiki page content language: en
wiki page sort key: Cocentral implies right-quotient-transitively central factor

C-closed implies powering-invariant
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/C-2Dclosed_implies_powering-2Dinvariant
wiki namespace: 0
wiki page content language: en
wiki page sort key: C-closed implies powering-invariant

C-closed implies local powering-invariant
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/C-2Dclosed_implies_local_powering-2Dinvariant
wiki namespace: 0
wiki page content language: en
wiki page sort key: C-closed implies local powering-invariant

CEP implies every relatively normal subgroup is weakly closed
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/CEP_implies_every_relatively_normal_subgroup_is_weakly_closed
wiki namespace: 0
wiki page content language: en
wiki page sort key: CEP implies every relatively normal subgroup is weakly closed

Normalizer of pronormal implies abnormal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Normalizer_of_pronormal_implies_abnormal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Normalizer of pronormal implies abnormal

Verbal subgroup of abelian group implies divisibility-closed
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Verbal_subgroup_of_abelian_group_implies_divisibility-2Dclosed
wiki namespace: 0
wiki page content language: en
wiki page sort key: Verbal subgroup of abelian group implies divisibility-closed

Cofactorial automorphism-invariant implies left-transitively 2-subnormal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Cofactorial_automorphism-2Dinvariant_implies_left-2Dtransitively_2-2Dsubnormal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Cofactorial automorphism-invariant implies left-transitively 2-subnormal

Direct factor implies transitively normal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Direct_factor_implies_transitively_normal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Direct factor implies transitively normal

Normal of finite index implies completely divisibility-closed
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Normal_of_finite_index_implies_completely_divisibility-2Dclosed
wiki namespace: 0
wiki page content language: en
wiki page sort key: Normal of finite index implies completely divisibility-closed

Every normal subgroup satisfies the quotient-to-subgroup powering-invariance implication
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Every_normal_subgroup_satisfies_the_quotient-2Dto-2Dsubgroup_powering-2Dinvariance_implication
wiki namespace: 0
wiki page content language: en
wiki page sort key: Every normal subgroup satisfies the quotient-to-subgroup powering-invariance implication

Fixed-point subgroup of a subgroup of the automorphism group implies local powering-invariant
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Fixed-2Dpoint_subgroup_of_a_subgroup_of_the_automorphism_group_implies_local_powering-2Dinvariant
wiki namespace: 0
wiki page content language: en
wiki page sort key: Fixed-point subgroup of a subgroup of the automorphism group implies local powering-invariant

Sylow implies WNSCDIN
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Sylow_implies_WNSCDIN
wiki namespace: 0
wiki page content language: en
wiki page sort key: Sylow implies WNSCDIN

Sylow implies isomorph-conjugate
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Sylow_implies_isomorph-2Dconjugate
wiki namespace: 0
wiki page content language: en
wiki page sort key: Sylow implies isomorph-conjugate

Intermediately isomorph-conjugate of normal implies pronormal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Intermediately_isomorph-2Dconjugate_of_normal_implies_pronormal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Intermediately isomorph-conjugate of normal implies pronormal

Marginal implies unconditionally closed
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Marginal_implies_unconditionally_closed
wiki namespace: 0
wiki page content language: en
wiki page sort key: Marginal implies unconditionally closed

Characteristic subgroup of abelian group is quotient-powering-invariant
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Characteristic_subgroup_of_abelian_group_is_quotient-2Dpowering-2Dinvariant
wiki namespace: 0
wiki page content language: en
wiki page sort key: Characteristic subgroup of abelian group is quotient-powering-invariant

Potentially verbal implies normal
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Potentially_verbal_implies_normal
wiki namespace: 0
wiki page content language: en
wiki page sort key: Potentially verbal implies normal

Supersolvable implies every nontrivial normal subgroup contains a cyclic normal subgroup
type: Subject
type: Subgroup 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

Finite index implies powering-invariant
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Finite_index_implies_powering-2Dinvariant
wiki namespace: 0
wiki page content language: en
wiki page sort key: Finite index implies powering-invariant

Finitely generated implies every subgroup of finite index has finitely many automorphic subgroups
type: Subject
type: Subgroup 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

Sylow implies order-dominated
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Sylow_implies_order-2Ddominated
wiki namespace: 0
wiki page content language: en
wiki page sort key: Sylow implies order-dominated

Fully invariant implies finite direct power-closed characteristic
type: Subject
type: Subgroup property implications
page: https://groupprops.subwiki.org/wiki/Fully_invariant_implies_finite_direct_power-2Dclosed_characteristic
wiki namespace: 0
wiki page content language: en
wiki page sort key: Fully invariant implies finite direct power-closed characteristic

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