]> 2026-06-11T16:36:44+00:00 Subgroup property implications 14 en 2008-05-07T22:39:22Z 2454594.4440046 Subgroup property implications WNSCDIN implies every normalizer-relatively normal conjugation-invariantly relatively normal subgroup is weakly closed 0 en WNSCDIN implies every normalizer-relatively normal conjugation-invariantly relatively normal subgroup is weakly closed Characteristic implies normal 0 en Characteristic implies normal Sylow normalizer implies weakly abnormal 0 en Sylow normalizer implies weakly abnormal Subgroup of index two is normal 0 en Subgroup of index two is normal Central factor implies transitively normal 0 en Central factor implies transitively normal Normal Hall implies permutably complemented 0 en Normal Hall implies permutably complemented Intermediately automorph-conjugate of normal implies weakly pronormal 0 en Intermediately automorph-conjugate of normal implies weakly pronormal Characteristic upper-hook AEP implies characteristic 0 en Characteristic upper-hook AEP implies characteristic Normal and self-centralizing implies coprime automorphism-faithful 0 en Normal and self-centralizing implies coprime automorphism-faithful Prime power order implies center is normality-large 0 en Prime power order implies center is normality-large Artinian implies periodic 0 en Artinian implies periodic Conjugate-join-closed subnormal implies join-transitively subnormal 0 en Conjugate-join-closed subnormal implies join-transitively subnormal Amalgam-characteristic implies potentially characteristic 0 en Amalgam-characteristic implies potentially characteristic Central implies normal 0 en Central implies normal Permutable implies locally subnormal 0 en Permutable implies locally subnormal Fully invariant implies characteristic 0 en Fully invariant implies characteristic Pronormal implies WNSCDIN 0 en Pronormal implies WNSCDIN Maximal among abelian normal implies self-centralizing in nilpotent 0 en Maximal among abelian normal implies self-centralizing in nilpotent Double coset index two implies maximal 0 en Double coset index two implies maximal Subgroup of finite index has a left transversal that is also a right transversal 0 en 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 0 en Subgroup of finite group has a left transversal that is also a right transversal Central factor implies normal 0 en Central factor implies normal Commutator of a transitively normal subgroup and a subset implies normal 0 en Commutator of a transitively normal subgroup and a subset implies normal Left-transitively complemented normal implies characteristic 0 en Left-transitively complemented normal implies characteristic Pronormal implies MWNSCDIN 0 en Pronormal implies MWNSCDIN Base of a wreath product implies right-transitively conjugate-permutable 0 en Base of a wreath product implies right-transitively conjugate-permutable Pronormal implies self-conjugate-permutable 0 en Pronormal implies self-conjugate-permutable Monolith is strictly characteristic 0 en Monolith is strictly characteristic Direct factor implies normal 0 en Direct factor implies normal Central implies normal satisfying the subgroup-to-quotient powering-invariance implication 0 en Central implies normal satisfying the subgroup-to-quotient powering-invariance implication Central implies image-potentially characteristic 0 en Central implies image-potentially characteristic Central implies amalgam-characteristic 0 en Central implies amalgam-characteristic Amalgam-characteristic implies image-potentially characteristic 0 en Amalgam-characteristic implies image-potentially characteristic Finite normal implies amalgam-characteristic 0 en Finite normal implies amalgam-characteristic Verbal implies fully invariant 0 en Verbal implies fully invariant Direct factor implies join-transitively central factor 0 en Direct factor implies join-transitively central factor Direct factor implies right-quotient-transitively central factor 0 en Direct factor implies right-quotient-transitively central factor Right-quotient-transitively central factor implies join-transitively central factor 0 en Right-quotient-transitively central factor implies join-transitively central factor Normal implies permutable 0 en Normal implies permutable Iterated agemo implies verbal 0 en Iterated agemo implies verbal Endomorphism kernel implies quotient-powering-invariant 0 en Endomorphism kernel implies quotient-powering-invariant Endomorphism image implies powering-invariant 0 en Endomorphism image implies powering-invariant Endomorphism image implies divisibility-closed 0 en Endomorphism image implies divisibility-closed Divisibility-closed implies powering-invariant 0 en Divisibility-closed implies powering-invariant Finite normal implies image-potentially characteristic 0 en Finite normal implies image-potentially characteristic Hall retract implies order-conjugate 0 en Hall retract implies order-conjugate Central implies potentially characteristic 0 en Central implies potentially characteristic Normal and centralizer-free implies automorphism-faithful 0 en Normal and centralizer-free implies automorphism-faithful Cocentral implies central factor 0 en Cocentral implies central factor 3-subnormal implies finite-conjugate-join-closed subnormal 0 en 3-subnormal implies finite-conjugate-join-closed subnormal 2-subnormal implies join-transitively subnormal 0 en 2-subnormal implies join-transitively subnormal 2-subnormal implies conjugate-permutable 0 en 2-subnormal implies conjugate-permutable Perfect subnormal implies join-transitively subnormal 0 en Perfect subnormal implies join-transitively subnormal Perfect subnormal implies subnormal-permutable 0 en Perfect subnormal implies subnormal-permutable Self-centralizing and minimal normal implies strictly characteristic 0 en Self-centralizing and minimal normal implies strictly characteristic Direct factor implies central factor 0 en Direct factor implies central factor Join of finitely many direct factors implies central factor 0 en Join of finitely many direct factors implies central factor Characteristic central factor of WNSCDIN implies WNSCDIN 0 en Characteristic central factor of WNSCDIN implies WNSCDIN Paranormal implies weakly normal 0 en Paranormal implies weakly normal Paranormal implies polynormal 0 en Paranormal implies polynormal Permutable implies ascendant 0 en Permutable implies ascendant Pronormal and subnormal implies normal 0 en Pronormal and subnormal implies normal Pronormal implies intermediately subnormal-to-normal 0 en Pronormal implies intermediately subnormal-to-normal Subgroup-cofactorial automorphism-invariant implies left-transitively 2-subnormal 0 en Subgroup-cofactorial automorphism-invariant implies left-transitively 2-subnormal Complemented normal implies endomorphism kernel 0 en Complemented normal implies endomorphism kernel Maximal implies pronormal 0 en Maximal implies pronormal Join with any distinct conjugate is the whole group implies pronormal 0 en Join with any distinct conjugate is the whole group implies pronormal Central implies potentially verbal in finite 0 en Central implies potentially verbal in finite Periodic normal implies potentially characteristic 0 en Periodic normal implies potentially characteristic Periodic normal implies amalgam-characteristic 0 en Periodic normal implies amalgam-characteristic Retract implies local divisibility-invariant 0 en Retract implies local divisibility-invariant Nilpotent implies every maximal subgroup is normal 0 en Nilpotent implies every maximal subgroup is normal Fully invariant direct factor implies left-transitively homomorph-containing 0 en Fully invariant direct factor implies left-transitively homomorph-containing Central subgroup implies join-transitively central factor 0 en Central subgroup implies join-transitively central factor Cyclic-quotient characteristic implies upward-closed characteristic 0 en Cyclic-quotient characteristic implies upward-closed characteristic Cyclic characteristic implies hereditarily characteristic 0 en Cyclic characteristic implies hereditarily characteristic Cocentral implies centralizer-dense 0 en Cocentral implies centralizer-dense Maximal implies modular 0 en Maximal implies modular Normal and self-centralizing implies normality-large 0 en Normal and self-centralizing implies normality-large NE implies weakly normal 0 en NE implies weakly normal Isomorph-containing implies characteristic 0 en Isomorph-containing implies characteristic Strongly paranormal implies paranormal 0 en Strongly paranormal implies paranormal Commutator of a normal subgroup and a subset implies 2-subnormal 0 en Commutator of a normal subgroup and a subset implies 2-subnormal Normalizer of intermediately subnormal-to-normal implies self-normalizing 0 en Normalizer of intermediately subnormal-to-normal implies self-normalizing Left-transitively permutable implies characteristic 0 en Left-transitively permutable implies characteristic Characteristic of CDIN implies CDIN 0 en Characteristic of CDIN implies CDIN Normal of finite index implies quotient-powering-invariant 0 en Normal of finite index implies quotient-powering-invariant Cyclic normal implies hereditarily normal 0 en Cyclic normal implies hereditarily normal Fully invariant of strictly characteristic implies strictly characteristic 0 en Fully invariant of strictly characteristic implies strictly characteristic Intersection of DPICF and direct factor is central factor 0 en Intersection of DPICF and direct factor is central factor Monolith is characteristic 0 en Monolith is characteristic Retract implies central factor of normalizer 0 en Retract implies central factor of normalizer Subgroup of finite abelian group implies finite-abelian-pi-potentially verbal 0 en Subgroup of finite abelian group implies finite-abelian-pi-potentially verbal Hall-extensible implies class-preserving 0 en Hall-extensible implies class-preserving Center of pronormal implies SCDIN 0 en Center of pronormal implies SCDIN Abelian and pronormal implies SCDIN 0 en Abelian and pronormal implies SCDIN Finite normal implies quotient-powering-invariant 0 en Finite normal implies quotient-powering-invariant Commutator of a group and a subgroup implies normal 0 en Commutator of a group and a subgroup implies normal Potentially characteristic implies normal 0 en Potentially characteristic implies normal Prehomomorph-contained implies strictly characteristic 0 en Prehomomorph-contained implies strictly characteristic Polynormal implies intermediately subnormal-to-normal 0 en Polynormal implies intermediately subnormal-to-normal Identity functor controls strong fusion for abelian Sylow subgroup 0 en Identity functor controls strong fusion for abelian Sylow subgroup Sylow implies MWNSCDIN 0 en Sylow implies MWNSCDIN Hall implies paracharacteristic 0 en Hall implies paracharacteristic Hall implies join of Sylow subgroups 0 en Hall implies join of Sylow subgroups Characteristic subgroup of abelian group implies powering-invariant 0 en Characteristic subgroup of abelian group implies powering-invariant Finite implies powering-invariant 0 en Finite implies powering-invariant Permutable implies modular 0 en Permutable implies modular Focal subgroup theorem 0 en Focal subgroup theorem Base of a wreath product implies right-transitively 2-subnormal 0 en Base of a wreath product implies right-transitively 2-subnormal Retract implies derived subgroup equals intersection with whole derived subgroup 0 en Retract implies derived subgroup equals intersection with whole derived subgroup Kernel of a characteristic action on an abelian group implies strongly image-potentially characteristic 0 en Kernel of a characteristic action on an abelian group implies strongly image-potentially characteristic Index four implies 2-subnormal or double coset index two 0 en Index four implies 2-subnormal or double coset index two Pronormal implies weakly normal 0 en Pronormal implies weakly normal Marginal implies direct power-closed characteristic 0 en Marginal implies direct power-closed characteristic Left-transitively 2-subnormal implies normal of characteristic 0 en Left-transitively 2-subnormal implies normal of characteristic Normal-homomorph-containing implies strictly characteristic 0 en Normal-homomorph-containing implies strictly characteristic Periodic normal implies image-potentially characteristic 0 en Periodic normal implies image-potentially characteristic Sylow implies order-conjugate 0 en Sylow implies order-conjugate Normality-preserving endomorphism-invariant implies finite direct power-closed characteristic 0 en Normality-preserving endomorphism-invariant implies finite direct power-closed characteristic Normality-preserving endomorphism-invariant implies characteristic 0 en Normality-preserving endomorphism-invariant implies characteristic Sylow and TI implies CDIN 0 en Sylow and TI implies CDIN CDIN of conjugacy-closed implies CDIN 0 en CDIN of conjugacy-closed implies CDIN Base of a wreath product implies subset-conjugacy-closed 0 en Base of a wreath product implies subset-conjugacy-closed SCDIN of subset-conjugacy-closed implies SCDIN 0 en SCDIN of subset-conjugacy-closed implies SCDIN Finite index implies completely divisibility-closed 0 en Finite index implies completely divisibility-closed Nilpotent-quotient implies subgroup-to-quotient powering-invariance implication 0 en Nilpotent-quotient implies subgroup-to-quotient powering-invariance implication Self-centralizing and minimal normal implies characteristic 0 en Self-centralizing and minimal normal implies characteristic Finite implies local powering-invariant 0 en Finite implies local powering-invariant Normal subgroup of nilpotent group satisfies the subgroup-to-quotient powering-invariance implication 0 en 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 0 en Normal subgroup contained in the hypercenter satisfies the subgroup-to-quotient powering-invariance implication Intersection of kernels of bihomomorphisms implies completely divisibility-closed 0 en Intersection of kernels of bihomomorphisms implies completely divisibility-closed Kernel of a bihomomorphism implies completely divisibility-closed 0 en Kernel of a bihomomorphism implies completely divisibility-closed Characteristic and self-centralizing implies coprime automorphism-faithful 0 en Characteristic and self-centralizing implies coprime automorphism-faithful Characteristic subgroup of abelian group implies intermediately powering-invariant 0 en Characteristic subgroup of abelian group implies intermediately powering-invariant 2-subnormal implies conjugate-join-closed subnormal 0 en 2-subnormal implies conjugate-join-closed subnormal Index three implies normal or double coset index two 0 en Index three implies normal or double coset index two Subhomomorph-containing implies right-transitively homomorph-containing 0 en Subhomomorph-containing implies right-transitively homomorph-containing Central implies abelian normal 0 en Central implies abelian normal Conjugacy-closed implies focal subgroup equals derived subgroup 0 en Conjugacy-closed implies focal subgroup equals derived subgroup Sylow implies automorph-conjugate 0 en Sylow implies automorph-conjugate Intermediately normal-to-characteristic implies intermediately subnormal-to-normal 0 en Intermediately normal-to-characteristic implies intermediately subnormal-to-normal Finite solvable-extensible implies class-preserving 0 en Finite solvable-extensible implies class-preserving Normal implies join-transitively subnormal 0 en Normal implies join-transitively subnormal Permutable and subnormal implies join-transitively subnormal 0 en Permutable and subnormal implies join-transitively subnormal Normal-potentially characteristic implies normal-extensible automorphism-invariant 0 en Normal-potentially characteristic implies normal-extensible automorphism-invariant Free factor implies self-normalizing or trivial 0 en Free factor implies self-normalizing or trivial Sylow and TI implies SCDIN 0 en Sylow and TI implies SCDIN Abnormal implies WNSCC 0 en Abnormal implies WNSCC Characteristic implies automorph-conjugate 0 en Characteristic implies automorph-conjugate Kernel of a bihomomorphism implies abelian-quotient 0 en Kernel of a bihomomorphism implies abelian-quotient Cocentral implies right-quotient-transitively central factor 0 en Cocentral implies right-quotient-transitively central factor C-closed implies powering-invariant 0 en C-closed implies powering-invariant C-closed implies local powering-invariant 0 en C-closed implies local powering-invariant CEP implies every relatively normal subgroup is weakly closed 0 en CEP implies every relatively normal subgroup is weakly closed Normalizer of pronormal implies abnormal 0 en Normalizer of pronormal implies abnormal Verbal subgroup of abelian group implies divisibility-closed 0 en Verbal subgroup of abelian group implies divisibility-closed Cofactorial automorphism-invariant implies left-transitively 2-subnormal 0 en Cofactorial automorphism-invariant implies left-transitively 2-subnormal Direct factor implies transitively normal 0 en Direct factor implies transitively normal Normal of finite index implies completely divisibility-closed 0 en Normal of finite index implies completely divisibility-closed Every normal subgroup satisfies the quotient-to-subgroup powering-invariance implication 0 en 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 0 en Fixed-point subgroup of a subgroup of the automorphism group implies local powering-invariant Sylow implies WNSCDIN 0 en Sylow implies WNSCDIN Sylow implies isomorph-conjugate 0 en Sylow implies isomorph-conjugate Intermediately isomorph-conjugate of normal implies pronormal 0 en Intermediately isomorph-conjugate of normal implies pronormal Marginal implies unconditionally closed 0 en Marginal implies unconditionally closed Characteristic subgroup of abelian group is quotient-powering-invariant 0 en Characteristic subgroup of abelian group is quotient-powering-invariant Potentially verbal implies normal 0 en Potentially verbal implies normal Supersolvable implies every nontrivial normal subgroup contains a cyclic normal subgroup 0 en Supersolvable implies every nontrivial normal subgroup contains a cyclic normal subgroup Finite index implies powering-invariant 0 en Finite index implies powering-invariant Finitely generated implies every subgroup of finite index has finitely many automorphic subgroups 0 en Finitely generated implies every subgroup of finite index has finitely many automorphic subgroups Sylow implies order-dominated 0 en Sylow implies order-dominated Fully invariant implies finite direct power-closed characteristic 0 en Fully invariant implies finite direct power-closed characteristic WNSCDIN implies every normalizer-relatively normal conjugation-invariantly relatively normal subgroup is weakly closed 0 en WNSCDIN implies every normalizer-relatively normal conjugation-invariantly relatively normal subgroup is weakly closed Characteristic implies normal 0 en Characteristic implies normal Sylow normalizer implies weakly abnormal 0 en Sylow normalizer implies weakly abnormal Subgroup of index two is normal 0 en Subgroup of index two is normal Central factor implies transitively normal 0 en Central factor implies transitively normal Normal Hall implies permutably complemented 0 en Normal Hall implies permutably complemented Intermediately automorph-conjugate of normal implies weakly pronormal 0 en Intermediately automorph-conjugate of normal implies weakly pronormal Characteristic upper-hook AEP implies characteristic 0 en Characteristic upper-hook AEP implies characteristic Normal and self-centralizing implies coprime automorphism-faithful 0 en Normal and self-centralizing implies coprime automorphism-faithful Prime power order implies center is normality-large 0 en Prime power order implies center is normality-large Artinian implies periodic 0 en Artinian implies periodic Conjugate-join-closed subnormal implies join-transitively subnormal 0 en Conjugate-join-closed subnormal implies join-transitively subnormal Amalgam-characteristic implies potentially characteristic 0 en Amalgam-characteristic implies potentially characteristic Central implies normal 0 en Central implies normal Permutable implies locally subnormal 0 en Permutable implies locally subnormal Fully invariant implies characteristic 0 en Fully invariant implies characteristic Pronormal implies WNSCDIN 0 en Pronormal implies WNSCDIN Maximal among abelian normal implies self-centralizing in nilpotent 0 en Maximal among abelian normal implies self-centralizing in nilpotent Double coset index two implies maximal 0 en Double coset index two implies maximal Subgroup of finite index has a left transversal that is also a right transversal 0 en 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 0 en Subgroup of finite group has a left transversal that is also a right transversal Central factor implies normal 0 en Central factor implies normal Commutator of a transitively normal subgroup and a subset implies normal 0 en Commutator of a transitively normal subgroup and a subset implies normal Left-transitively complemented normal implies characteristic 0 en Left-transitively complemented normal implies characteristic Pronormal implies MWNSCDIN 0 en Pronormal implies MWNSCDIN Base of a wreath product implies right-transitively conjugate-permutable 0 en Base of a wreath product implies right-transitively conjugate-permutable Pronormal implies self-conjugate-permutable 0 en Pronormal implies self-conjugate-permutable Monolith is strictly characteristic 0 en Monolith is strictly characteristic Direct factor implies normal 0 en Direct factor implies normal Central implies normal satisfying the subgroup-to-quotient powering-invariance implication 0 en Central implies normal satisfying the subgroup-to-quotient powering-invariance implication Central implies image-potentially characteristic 0 en Central implies image-potentially characteristic Central implies amalgam-characteristic 0 en Central implies amalgam-characteristic Amalgam-characteristic implies image-potentially characteristic 0 en Amalgam-characteristic implies image-potentially characteristic Finite normal implies amalgam-characteristic 0 en Finite normal implies amalgam-characteristic Verbal implies fully invariant 0 en Verbal implies fully invariant Direct factor implies join-transitively central factor 0 en Direct factor implies join-transitively central factor Direct factor implies right-quotient-transitively central factor 0 en Direct factor implies right-quotient-transitively central factor Right-quotient-transitively central factor implies join-transitively central factor 0 en Right-quotient-transitively central factor implies join-transitively central factor Normal implies permutable 0 en Normal implies permutable Iterated agemo implies verbal 0 en Iterated agemo implies verbal Endomorphism kernel implies quotient-powering-invariant 0 en Endomorphism kernel implies quotient-powering-invariant Endomorphism image implies powering-invariant 0 en Endomorphism image implies powering-invariant Endomorphism image implies divisibility-closed 0 en Endomorphism image implies divisibility-closed Divisibility-closed implies powering-invariant 0 en Divisibility-closed implies powering-invariant Finite normal implies image-potentially characteristic 0 en Finite normal implies image-potentially characteristic Hall retract implies order-conjugate 0 en Hall retract implies order-conjugate Central implies potentially characteristic 0 en Central implies potentially characteristic Normal and centralizer-free implies automorphism-faithful 0 en Normal and centralizer-free implies automorphism-faithful Cocentral implies central factor 0 en Cocentral implies central factor 3-subnormal implies finite-conjugate-join-closed subnormal 0 en 3-subnormal implies finite-conjugate-join-closed subnormal 2-subnormal implies join-transitively subnormal 0 en 2-subnormal implies join-transitively subnormal 2-subnormal implies conjugate-permutable 0 en 2-subnormal implies conjugate-permutable Perfect subnormal implies join-transitively subnormal 0 en Perfect subnormal implies join-transitively subnormal Perfect subnormal implies subnormal-permutable 0 en Perfect subnormal implies subnormal-permutable Self-centralizing and minimal normal implies strictly characteristic 0 en Self-centralizing and minimal normal implies strictly characteristic Direct factor implies central factor 0 en Direct factor implies central factor Join of finitely many direct factors implies central factor 0 en Join of finitely many direct factors implies central factor Characteristic central factor of WNSCDIN implies WNSCDIN 0 en Characteristic central factor of WNSCDIN implies WNSCDIN Paranormal implies weakly normal 0 en Paranormal implies weakly normal Paranormal implies polynormal 0 en Paranormal implies polynormal Permutable implies ascendant 0 en Permutable implies ascendant Pronormal and subnormal implies normal 0 en Pronormal and subnormal implies normal Pronormal implies intermediately subnormal-to-normal 0 en Pronormal implies intermediately subnormal-to-normal Subgroup-cofactorial automorphism-invariant implies left-transitively 2-subnormal 0 en Subgroup-cofactorial automorphism-invariant implies left-transitively 2-subnormal Complemented normal implies endomorphism kernel 0 en Complemented normal implies endomorphism kernel Maximal implies pronormal 0 en Maximal implies pronormal Join with any distinct conjugate is the whole group implies pronormal 0 en Join with any distinct conjugate is the whole group implies pronormal Central implies potentially verbal in finite 0 en Central implies potentially verbal in finite Periodic normal implies potentially characteristic 0 en Periodic normal implies potentially characteristic Periodic normal implies amalgam-characteristic 0 en Periodic normal implies amalgam-characteristic Retract implies local divisibility-invariant 0 en Retract implies local divisibility-invariant Nilpotent implies every maximal subgroup is normal 0 en Nilpotent implies every maximal subgroup is normal Fully invariant direct factor implies left-transitively homomorph-containing 0 en Fully invariant direct factor implies left-transitively homomorph-containing Central subgroup implies join-transitively central factor 0 en Central subgroup implies join-transitively central factor Cyclic-quotient characteristic implies upward-closed characteristic 0 en Cyclic-quotient characteristic implies upward-closed characteristic Cyclic characteristic implies hereditarily characteristic 0 en Cyclic characteristic implies hereditarily characteristic Cocentral implies centralizer-dense 0 en Cocentral implies centralizer-dense Maximal implies modular 0 en Maximal implies modular Normal and self-centralizing implies normality-large 0 en Normal and self-centralizing implies normality-large NE implies weakly normal 0 en NE implies weakly normal Isomorph-containing implies characteristic 0 en Isomorph-containing implies characteristic Strongly paranormal implies paranormal 0 en Strongly paranormal implies paranormal Commutator of a normal subgroup and a subset implies 2-subnormal 0 en Commutator of a normal subgroup and a subset implies 2-subnormal Normalizer of intermediately subnormal-to-normal implies self-normalizing 0 en Normalizer of intermediately subnormal-to-normal implies self-normalizing Left-transitively permutable implies characteristic 0 en Left-transitively permutable implies characteristic Characteristic of CDIN implies CDIN 0 en Characteristic of CDIN implies CDIN Normal of finite index implies quotient-powering-invariant 0 en Normal of finite index implies quotient-powering-invariant Cyclic normal implies hereditarily normal 0 en Cyclic normal implies hereditarily normal Fully invariant of strictly characteristic implies strictly characteristic 0 en Fully invariant of strictly characteristic implies strictly characteristic Intersection of DPICF and direct factor is central factor 0 en Intersection of DPICF and direct factor is central factor Monolith is characteristic 0 en Monolith is characteristic Retract implies central factor of normalizer 0 en Retract implies central factor of normalizer Subgroup of finite abelian group implies finite-abelian-pi-potentially verbal 0 en Subgroup of finite abelian group implies finite-abelian-pi-potentially verbal Hall-extensible implies class-preserving 0 en Hall-extensible implies class-preserving Center of pronormal implies SCDIN 0 en Center of pronormal implies SCDIN Abelian and pronormal implies SCDIN 0 en Abelian and pronormal implies SCDIN Finite normal implies quotient-powering-invariant 0 en Finite normal implies quotient-powering-invariant Commutator of a group and a subgroup implies normal 0 en Commutator of a group and a subgroup implies normal Potentially characteristic implies normal 0 en Potentially characteristic implies normal Prehomomorph-contained implies strictly characteristic 0 en Prehomomorph-contained implies strictly characteristic INST 132 en INST