]> 2026-07-27T23:13:25+00:00 Subgroup property non-implications 14 en 2008-05-07T22:39:23Z 2454594.4440162 Subgroup property non-implications Characteristic not implies powering-invariant in nilpotent group 0 en Characteristic not implies powering-invariant in nilpotent group Normal not implies characteristic 0 en Normal not implies characteristic Descendant not implies subnormal 0 en Descendant not implies subnormal Ascendant not implies subnormal 0 en Ascendant not implies subnormal Normal not implies left-transitively fixed-depth subnormal 0 en Normal not implies left-transitively fixed-depth subnormal Normal not implies right-transitively fixed-depth subnormal 0 en Normal not implies right-transitively fixed-depth subnormal Nilpotent not implies ACIC 0 en Nilpotent not implies ACIC Hall not implies pronormal 0 en Hall not implies pronormal Hall not implies order-conjugate 0 en Hall not implies order-conjugate Normal not implies strongly potentially characteristic 0 en Normal not implies strongly potentially characteristic Normal not implies normal-potentially characteristic 0 en Normal not implies normal-potentially characteristic Hall not implies WNSCDIN 0 en Hall not implies WNSCDIN Pronormal not implies NE 0 en Pronormal not implies NE Sylow not implies NE 0 en Sylow not implies NE Subnormal not implies conjugate-permutable 0 en Subnormal not implies conjugate-permutable Order-conjugate and Hall not implies order-dominating 0 en Order-conjugate and Hall not implies order-dominating Left-transitively WNSCDIN not implies normal 0 en Left-transitively WNSCDIN not implies normal Left-transitively homomorph-containing not implies subhomomorph-containing 0 en Left-transitively homomorph-containing not implies subhomomorph-containing Strictly characteristic not implies fully invariant 0 en Strictly characteristic not implies fully invariant Locally inner automorphism-balanced not implies central factor 0 en Locally inner automorphism-balanced not implies central factor Conjugacy-closed normal not implies central factor 0 en Conjugacy-closed normal not implies central factor Permutable not implies normal 0 en Permutable not implies normal Potentially characteristic not implies characteristic-potentially characteristic 0 en Potentially characteristic not implies characteristic-potentially characteristic Potentially characteristic not implies normal-potentially characteristic 0 en Potentially characteristic not implies normal-potentially characteristic Potentially characteristic not implies normal-extensible automorphism-invariant 0 en Potentially characteristic not implies normal-extensible automorphism-invariant Verbal not implies commutator-verbal 0 en Verbal not implies commutator-verbal No common composition factor with quotient group not implies complemented 0 en No common composition factor with quotient group not implies complemented Subnormal not implies normal 0 en Subnormal not implies normal Automorph-permutable not implies permutable 0 en Automorph-permutable not implies permutable Conjugacy-closed and Hall not implies retract 0 en Conjugacy-closed and Hall not implies retract No nontrivial homomorphism to quotient group not implies complemented normal 0 en No nontrivial homomorphism to quotient group not implies complemented normal Subisomorph-containing not implies homomorph-containing 0 en Subisomorph-containing not implies homomorph-containing Cocentral not implies amalgam-characteristic 0 en Cocentral not implies amalgam-characteristic Characteristic not implies strictly characteristic 0 en Characteristic not implies strictly characteristic Order-conjugate and Hall not implies order-dominated 0 en Order-conjugate and Hall not implies order-dominated Subgroup of abelian group not implies abelian-extensible automorphism-invariant 0 en Subgroup of abelian group not implies abelian-extensible automorphism-invariant Weakly normal not implies NE 0 en Weakly normal not implies NE 2-subnormal not implies hypernormalized 0 en 2-subnormal not implies hypernormalized Quotient group need not be isomorphic to any subgroup 0 en Quotient group need not be isomorphic to any subgroup Normal not implies amalgam-characteristic 0 en Normal not implies amalgam-characteristic Normal not implies potentially verbal 0 en Normal not implies potentially verbal Characteristic not implies potentially fully invariant 0 en Characteristic not implies potentially fully invariant Normal not implies potentially fully invariant 0 en Normal not implies potentially fully invariant Characteristic not implies amalgam-characteristic 0 en Characteristic not implies amalgam-characteristic Characteristic not implies fully invariant 0 en Characteristic not implies fully invariant Iterated agemo subgroup not implies agemo subgroup 0 en Iterated agemo subgroup not implies agemo subgroup Image-closed characteristic not implies fully invariant 0 en Image-closed characteristic not implies fully invariant Abnormal normalizer not implies pronormal 0 en Abnormal normalizer not implies pronormal Fusion system-relatively weakly closed not implies isomorph-normal 0 en Fusion system-relatively weakly closed not implies isomorph-normal 2-subnormal not implies automorph-permutable 0 en 2-subnormal not implies automorph-permutable Finite direct power-closed characteristic not implies fully invariant 0 en Finite direct power-closed characteristic not implies fully invariant Right-transitively homomorph-containing not implies subhomomorph-containing 0 en Right-transitively homomorph-containing not implies subhomomorph-containing Complemented central factor not implies direct factor 0 en Complemented central factor not implies direct factor Normal-extensible not implies inner 0 en Normal-extensible not implies inner Finite not implies divisibility-closed in abelian group 0 en Finite not implies divisibility-closed in abelian group Characteristic not implies quasiautomorphism-invariant 0 en Characteristic not implies quasiautomorphism-invariant Finite solvable not implies p-normal 0 en Finite solvable not implies p-normal Homomorph-containing not implies no nontrivial homomorphism to quotient group 0 en Homomorph-containing not implies no nontrivial homomorphism to quotient group Transitively normal not implies central factor 0 en Transitively normal not implies central factor Local powering-invariant not implies local divisibility-closed 0 en Local powering-invariant not implies local divisibility-closed Normal not implies normal-potentially relatively characteristic 0 en Normal not implies normal-potentially relatively characteristic Characteristic not implies injective endomorphism-invariant 0 en Characteristic not implies injective endomorphism-invariant Normal-potentially characteristic not implies characteristic 0 en Normal-potentially characteristic not implies characteristic Normal not implies central factor 0 en Normal not implies central factor Sylow not implies local divisibility-closed 0 en Sylow not implies local divisibility-closed Characteristic-isomorph-free not implies normal-isomorph-free in finite 0 en Characteristic-isomorph-free not implies normal-isomorph-free in finite Endomorphism kernel not implies complemented normal 0 en Endomorphism kernel not implies complemented normal Fully invariant subgroup of abelian group not implies divisibility-closed 0 en Fully invariant subgroup of abelian group not implies divisibility-closed Sylow not implies CDIN 0 en Sylow not implies CDIN Powering-invariance does not satisfy lower central series condition in nilpotent group 0 en Powering-invariance does not satisfy lower central series condition in nilpotent group Subgroup of abelian group not implies abelian-potentially characteristic 0 en Subgroup of abelian group not implies abelian-potentially characteristic Normal not implies image-potentially fully invariant 0 en Normal not implies image-potentially fully invariant Conjugate-comparable not implies normal 0 en Conjugate-comparable not implies normal Characteristic subgroup of abelian group not implies local powering-invariant 0 en Characteristic subgroup of abelian group not implies local powering-invariant Powering-invariant not implies local powering-invariant 0 en Powering-invariant not implies local powering-invariant Lattice-complemented not implies permutably complemented 0 en Lattice-complemented not implies permutably complemented Verbal subgroup of abelian group not implies local powering-invariant 0 en Verbal subgroup of abelian group not implies local powering-invariant Order-conjugate not implies order-dominating 0 en Order-conjugate not implies order-dominating Normal-extensible not implies normal 0 en Normal-extensible not implies normal Verbal not implies iterated agemo 0 en Verbal not implies iterated agemo Central factor not implies direct factor 0 en Central factor not implies direct factor Normal-isomorph-free not implies isomorph-free in finite 0 en Normal-isomorph-free not implies isomorph-free in finite Image-closed fully invariant not implies verbal 0 en Image-closed fully invariant not implies verbal 3-subnormal not implies finite-automorph-join-closed subnormal 0 en 3-subnormal not implies finite-automorph-join-closed subnormal Characteristic subgroup of abelian group not implies divisibility-closed 0 en Characteristic subgroup of abelian group not implies divisibility-closed Direct factor not implies characteristic 0 en Direct factor not implies characteristic Paranormal not implies pronormal 0 en Paranormal not implies pronormal Abelian normal not implies central 0 en Abelian normal not implies central Contranormal not implies self-normalizing 0 en Contranormal not implies self-normalizing Normal not implies direct factor 0 en Normal not implies direct factor Complemented characteristic not implies left-transitively complemented normal 0 en Complemented characteristic not implies left-transitively complemented normal Divisibility-closed not implies local divisibility-closed 0 en Divisibility-closed not implies local divisibility-closed Index two not implies characteristic 0 en Index two not implies characteristic Existentially bound-word not implies verbal 0 en Existentially bound-word not implies verbal Complemented normal not implies direct factor 0 en Complemented normal not implies direct factor P-automorphism-invariant not implies characteristic 0 en P-automorphism-invariant not implies characteristic Characteristic direct factor not implies fully invariant 0 en Characteristic direct factor not implies fully invariant Powering-invariant not implies divisibility-closed 0 en Powering-invariant not implies divisibility-closed Endomorph-dominating not implies automorph-conjugate 0 en Endomorph-dominating not implies automorph-conjugate Finite index not implies local powering-invariant 0 en Finite index not implies local powering-invariant 1-closed transversal not implies permutably complemented 0 en 1-closed transversal not implies permutably complemented Order-conjugate not implies order-dominated 0 en Order-conjugate not implies order-dominated Characteristic not implies characteristic-isomorph-free in finite 0 en Characteristic not implies characteristic-isomorph-free in finite Subhomomorph-containing not implies variety-containing 0 en Subhomomorph-containing not implies variety-containing Subgroup of abelian group not implies powering-invariant 0 en Subgroup of abelian group not implies powering-invariant Characteristic not implies elementarily characteristic 0 en Characteristic not implies elementarily characteristic Powering-invariant and normal not implies quotient-powering-invariant 0 en Powering-invariant and normal not implies quotient-powering-invariant Base of a wreath product not implies elliptic 0 en Base of a wreath product not implies elliptic Conjugacy-closed not implies weak subset-conjugacy-closed 0 en Conjugacy-closed not implies weak subset-conjugacy-closed P-normal not implies p-solvable 0 en P-normal not implies p-solvable Self-normalizing not implies contranormal 0 en Self-normalizing not implies contranormal No nontrivial homomorphism from quotient group not implies characteristic 0 en No nontrivial homomorphism from quotient group not implies characteristic Left-transitively WNSCDIN not implies characteristic 0 en Left-transitively WNSCDIN not implies characteristic Series-equivalent characteristic subgroups may be distinct 0 en Series-equivalent characteristic subgroups may be distinct Pure subgroup of torsion-free abelian group not implies direct factor 0 en Pure subgroup of torsion-free abelian group not implies direct factor Direct factor not implies amalgam-characteristic 0 en Direct factor not implies amalgam-characteristic Abelian-quotient not implies kernel of a bihomomorphism 0 en Abelian-quotient not implies kernel of a bihomomorphism Characteristic not implies direct factor 0 en Characteristic not implies direct factor Quasiautomorphism-invariant not implies 1-automorphism-invariant 0 en Quasiautomorphism-invariant not implies 1-automorphism-invariant Transitively normal not implies conjugacy-closed normal 0 en Transitively normal not implies conjugacy-closed normal Intermediately characteristic not implies isomorph-containing in group of prime power order 0 en Intermediately characteristic not implies isomorph-containing in group of prime power order Complemented normal not implies local powering-invariant 0 en Complemented normal not implies local powering-invariant 4-subnormal not implies finite-conjugate-join-closed subnormal 0 en 4-subnormal not implies finite-conjugate-join-closed subnormal Abelian-quotient not implies cocentral 0 en Abelian-quotient not implies cocentral Modular not implies permutable 0 en Modular not implies permutable Pronormal not implies join with any distinct conjugate is the whole group 0 en Pronormal not implies join with any distinct conjugate is the whole group Characteristic not implies powering-invariant in nilpotent group 0 en Characteristic not implies powering-invariant in nilpotent group Normal not implies characteristic 0 en Normal not implies characteristic Descendant not implies subnormal 0 en Descendant not implies subnormal Ascendant not implies subnormal 0 en Ascendant not implies subnormal Normal not implies left-transitively fixed-depth subnormal 0 en Normal not implies left-transitively fixed-depth subnormal Normal not implies right-transitively fixed-depth subnormal 0 en Normal not implies right-transitively fixed-depth subnormal Nilpotent not implies ACIC 0 en Nilpotent not implies ACIC Hall not implies pronormal 0 en Hall not implies pronormal Hall not implies order-conjugate 0 en Hall not implies order-conjugate Normal not implies strongly potentially characteristic 0 en Normal not implies strongly potentially characteristic Normal not implies normal-potentially characteristic 0 en Normal not implies normal-potentially characteristic Hall not implies WNSCDIN 0 en Hall not implies WNSCDIN Pronormal not implies NE 0 en Pronormal not implies NE Sylow not implies NE 0 en Sylow not implies NE Subnormal not implies conjugate-permutable 0 en Subnormal not implies conjugate-permutable Order-conjugate and Hall not implies order-dominating 0 en Order-conjugate and Hall not implies order-dominating Left-transitively WNSCDIN not implies normal 0 en Left-transitively WNSCDIN not implies normal Left-transitively homomorph-containing not implies subhomomorph-containing 0 en Left-transitively homomorph-containing not implies subhomomorph-containing Strictly characteristic not implies fully invariant 0 en Strictly characteristic not implies fully invariant Locally inner automorphism-balanced not implies central factor 0 en Locally inner automorphism-balanced not implies central factor Conjugacy-closed normal not implies central factor 0 en Conjugacy-closed normal not implies central factor Permutable not implies normal 0 en Permutable not implies normal Potentially characteristic not implies characteristic-potentially characteristic 0 en Potentially characteristic not implies characteristic-potentially characteristic Potentially characteristic not implies normal-potentially characteristic 0 en Potentially characteristic not implies normal-potentially characteristic Potentially characteristic not implies normal-extensible automorphism-invariant 0 en Potentially characteristic not implies normal-extensible automorphism-invariant Verbal not implies commutator-verbal 0 en Verbal not implies commutator-verbal No common composition factor with quotient group not implies complemented 0 en No common composition factor with quotient group not implies complemented Subnormal not implies normal 0 en Subnormal not implies normal Automorph-permutable not implies permutable 0 en Automorph-permutable not implies permutable Conjugacy-closed and Hall not implies retract 0 en Conjugacy-closed and Hall not implies retract No nontrivial homomorphism to quotient group not implies complemented normal 0 en No nontrivial homomorphism to quotient group not implies complemented normal Subisomorph-containing not implies homomorph-containing 0 en Subisomorph-containing not implies homomorph-containing Cocentral not implies amalgam-characteristic 0 en Cocentral not implies amalgam-characteristic Characteristic not implies strictly characteristic 0 en Characteristic not implies strictly characteristic Order-conjugate and Hall not implies order-dominated 0 en Order-conjugate and Hall not implies order-dominated Subgroup of abelian group not implies abelian-extensible automorphism-invariant 0 en Subgroup of abelian group not implies abelian-extensible automorphism-invariant Weakly normal not implies NE 0 en Weakly normal not implies NE 2-subnormal not implies hypernormalized 0 en 2-subnormal not implies hypernormalized Quotient group need not be isomorphic to any subgroup 0 en Quotient group need not be isomorphic to any subgroup Normal not implies amalgam-characteristic 0 en Normal not implies amalgam-characteristic Normal not implies potentially verbal 0 en Normal not implies potentially verbal Characteristic not implies potentially fully invariant 0 en Characteristic not implies potentially fully invariant Normal not implies potentially fully invariant 0 en Normal not implies potentially fully invariant Characteristic not implies amalgam-characteristic 0 en Characteristic not implies amalgam-characteristic Characteristic not implies fully invariant 0 en Characteristic not implies fully invariant Iterated agemo subgroup not implies agemo subgroup 0 en Iterated agemo subgroup not implies agemo subgroup Image-closed characteristic not implies fully invariant 0 en Image-closed characteristic not implies fully invariant Abnormal normalizer not implies pronormal 0 en Abnormal normalizer not implies pronormal Fusion system-relatively weakly closed not implies isomorph-normal 0 en Fusion system-relatively weakly closed not implies isomorph-normal 2-subnormal not implies automorph-permutable 0 en 2-subnormal not implies automorph-permutable Finite direct power-closed characteristic not implies fully invariant 0 en Finite direct power-closed characteristic not implies fully invariant Right-transitively homomorph-containing not implies subhomomorph-containing 0 en Right-transitively homomorph-containing not implies subhomomorph-containing Complemented central factor not implies direct factor 0 en Complemented central factor not implies direct factor Normal-extensible not implies inner 0 en Normal-extensible not implies inner Finite not implies divisibility-closed in abelian group 0 en Finite not implies divisibility-closed in abelian group Characteristic not implies quasiautomorphism-invariant 0 en Characteristic not implies quasiautomorphism-invariant Finite solvable not implies p-normal 0 en Finite solvable not implies p-normal Homomorph-containing not implies no nontrivial homomorphism to quotient group 0 en Homomorph-containing not implies no nontrivial homomorphism to quotient group Transitively normal not implies central factor 0 en Transitively normal not implies central factor Local powering-invariant not implies local divisibility-closed 0 en Local powering-invariant not implies local divisibility-closed Normal not implies normal-potentially relatively characteristic 0 en Normal not implies normal-potentially relatively characteristic Characteristic not implies injective endomorphism-invariant 0 en Characteristic not implies injective endomorphism-invariant Normal-potentially characteristic not implies characteristic 0 en Normal-potentially characteristic not implies characteristic Normal not implies central factor 0 en Normal not implies central factor Sylow not implies local divisibility-closed 0 en Sylow not implies local divisibility-closed Characteristic-isomorph-free not implies normal-isomorph-free in finite 0 en Characteristic-isomorph-free not implies normal-isomorph-free in finite Endomorphism kernel not implies complemented normal 0 en Endomorphism kernel not implies complemented normal Fully invariant subgroup of abelian group not implies divisibility-closed 0 en Fully invariant subgroup of abelian group not implies divisibility-closed Sylow not implies CDIN 0 en Sylow not implies CDIN Powering-invariance does not satisfy lower central series condition in nilpotent group 0 en Powering-invariance does not satisfy lower central series condition in nilpotent group Subgroup of abelian group not implies abelian-potentially characteristic 0 en Subgroup of abelian group not implies abelian-potentially characteristic Normal not implies image-potentially fully invariant 0 en Normal not implies image-potentially fully invariant Conjugate-comparable not implies normal 0 en Conjugate-comparable not implies normal Characteristic subgroup of abelian group not implies local powering-invariant 0 en Characteristic subgroup of abelian group not implies local powering-invariant Powering-invariant not implies local powering-invariant 0 en Powering-invariant not implies local powering-invariant Lattice-complemented not implies permutably complemented 0 en Lattice-complemented not implies permutably complemented Verbal subgroup of abelian group not implies local powering-invariant 0 en Verbal subgroup of abelian group not implies local powering-invariant Order-conjugate not implies order-dominating 0 en Order-conjugate not implies order-dominating Normal-extensible not implies normal 0 en Normal-extensible not implies normal Verbal not implies iterated agemo 0 en Verbal not implies iterated agemo Central factor not implies direct factor 0 en Central factor not implies direct factor Normal-isomorph-free not implies isomorph-free in finite 0 en Normal-isomorph-free not implies isomorph-free in finite Image-closed fully invariant not implies verbal 0 en Image-closed fully invariant not implies verbal 3-subnormal not implies finite-automorph-join-closed subnormal 0 en 3-subnormal not implies finite-automorph-join-closed subnormal Characteristic subgroup of abelian group not implies divisibility-closed 0 en Characteristic subgroup of abelian group not implies divisibility-closed Direct factor not implies characteristic 0 en Direct factor not implies characteristic Paranormal not implies pronormal 0 en Paranormal not implies pronormal Abelian normal not implies central 0 en Abelian normal not implies central Contranormal not implies self-normalizing 0 en Contranormal not implies self-normalizing Normal not implies direct factor 0 en Normal not implies direct factor Complemented characteristic not implies left-transitively complemented normal 0 en Complemented characteristic not implies left-transitively complemented normal Divisibility-closed not implies local divisibility-closed 0 en Divisibility-closed not implies local divisibility-closed Index two not implies characteristic 0 en Index two not implies characteristic Existentially bound-word not implies verbal 0 en Existentially bound-word not implies verbal Complemented normal not implies direct factor 0 en Complemented normal not implies direct factor P-automorphism-invariant not implies characteristic 0 en P-automorphism-invariant not implies characteristic Characteristic direct factor not implies fully invariant 0 en Characteristic direct factor not implies fully invariant Powering-invariant not implies divisibility-closed 0 en Powering-invariant not implies divisibility-closed Endomorph-dominating not implies automorph-conjugate 0 en Endomorph-dominating not implies automorph-conjugate Finite index not implies local powering-invariant 0 en Finite index not implies local powering-invariant INST 132 en INST