]> 2026-06-11T16:36:14+00:00 Group property implications 14 en Group property implications Dedekind implies ACIC 0 en Dedekind implies ACIC Prime power order implies nilpotent 0 en Prime power order implies nilpotent Nilpotent implies normalizer condition 0 en Nilpotent implies normalizer condition Cyclic over central implies abelian 0 en Cyclic over central implies abelian Every finite group has a finite composition series 0 en Every finite group has a finite composition series P-Frattini-realizable implies not non-abelian cyclic-center 0 en P-Frattini-realizable implies not non-abelian cyclic-center Order has only two prime factors implies solvable 0 en Order has only two prime factors implies solvable Dihedral implies solvable 0 en Dihedral implies solvable Monomial implies solvable 0 en Monomial implies solvable Supersolvable implies nilpotent derived subgroup 0 en Supersolvable implies nilpotent derived subgroup Abelian implies every element is automorphic to its inverse 0 en Abelian implies every element is automorphic to its inverse Finite minimal simple implies 2-generated 0 en Finite minimal simple implies 2-generated Odd-order implies solvable 0 en Odd-order implies solvable Finitely generated and residually finite implies Hopfian 0 en Finitely generated and residually finite implies Hopfian Finitely generated implies finitely many homomorphisms to any finite group 0 en Finitely generated implies finitely many homomorphisms to any finite group Linearly primitive implies cyclic-center 0 en Linearly primitive implies cyclic-center Ascending chain condition on normal subgroups implies Hopfian 0 en Ascending chain condition on normal subgroups implies Hopfian Ascending chain condition on subnormal subgroups implies subnormal join property 0 en Ascending chain condition on subnormal subgroups implies subnormal join property Frattini-embedded normal-realizable implies every automorph-conjugate subgroup is characteristic 0 en Frattini-embedded normal-realizable implies every automorph-conjugate subgroup is characteristic Strongly p-solvable implies Glauberman type for odd p 0 en Strongly p-solvable implies Glauberman type for odd p P-solvable implies p-constrained 0 en P-solvable implies p-constrained Maximal among abelian normal implies self-centralizing in nilpotent 0 en Maximal among abelian normal implies self-centralizing in nilpotent Finitely many homomorphisms to any finite group implies every subgroup of finite index has finitely many automorphic subgroups 0 en Finitely many homomorphisms to any finite group implies every subgroup of finite index has finitely many automorphic subgroups Finite implies subnormal join property 0 en Finite implies subnormal join property Automorphism group is transitive on non-identity elements implies characteristically simple 0 en Automorphism group is transitive on non-identity elements implies characteristically simple Finite simple implies 2-generated 0 en Finite simple implies 2-generated Glauberman type implies ZJ-functor controls fusion 0 en Glauberman type implies ZJ-functor controls fusion Finite normal implies amalgam-characteristic 0 en Finite normal implies amalgam-characteristic Conjugacy-separable implies residually finite 0 en Conjugacy-separable implies residually finite Cyclic quotient of automorphism group by class-preserving automorphism group implies same orbit sizes of conjugacy classes and irreducible representations under automorphism group 0 en Cyclic quotient of automorphism group by class-preserving automorphism group implies same orbit sizes of conjugacy classes and irreducible representations under automorphism group Nilpotent implies solvable 0 en Nilpotent implies solvable Nilpotent implies every subgroup is subnormal 0 en Nilpotent implies every subgroup is subnormal Every subgroup is subnormal implies normalizer condition 0 en Every subgroup is subnormal implies normalizer condition Schur multiplier of cyclic group is trivial 0 en Schur multiplier of cyclic group is trivial Class-inverting automorphism implies every element is automorphic to its inverse 0 en Class-inverting automorphism implies every element is automorphic to its inverse Cyclic implies abelian 0 en Cyclic implies abelian Nilpotent derived subgroup implies subnormal join property 0 en Nilpotent derived subgroup implies subnormal join property Nilpotent implies every nontrivial normal subgroup contains a cyclic normal subgroup 0 en Nilpotent implies every nontrivial normal subgroup contains a cyclic normal subgroup Nilpotent implies center is normality-large 0 en Nilpotent implies center is normality-large Center is normality-large implies every nontrivial normal subgroup contains a cyclic normal subgroup 0 en Center is normality-large implies every nontrivial normal subgroup contains a cyclic normal subgroup Metabelian implies subnormal join property 0 en Metabelian implies subnormal join property Finitely generated abelian is subgroup-closed 0 en Finitely generated abelian is subgroup-closed Frattini-embedded normal-realizable implies inner-in-automorphism-Frattini 0 en Frattini-embedded normal-realizable implies inner-in-automorphism-Frattini Extraspecial implies Camina 0 en Extraspecial implies Camina Prime power order implies subgroups of all orders dividing the group order 0 en Prime power order implies subgroups of all orders dividing the group order Schur multiplier of Z-group is trivial 0 en Schur multiplier of Z-group is trivial Nilpotent implies every maximal subgroup is normal 0 en Nilpotent implies every maximal subgroup is normal Normalizer condition implies every maximal subgroup is normal 0 en Normalizer condition implies every maximal subgroup is normal Multiplicative group of a field implies every finite subgroup is cyclic 0 en Multiplicative group of a field implies every finite subgroup is cyclic At most n elements of order dividing n implies every finite subgroup is cyclic 0 en At most n elements of order dividing n implies every finite subgroup is cyclic Free implies residually nilpotent 0 en Free implies residually nilpotent FZ implies generalized subnormal join property 0 en FZ implies generalized subnormal join property Abelian implies ACIC 0 en Abelian implies ACIC Locally cyclic implies abelian automorphism group 0 en Locally cyclic implies abelian automorphism group Artinian implies co-Hopfian 0 en Artinian implies co-Hopfian Algebra group implies power degree group for field size 0 en Algebra group implies power degree group for field size Free implies residually finite 0 en Free implies residually finite SQ-universal implies no nontrivial identity 0 en SQ-universal implies no nontrivial identity Additive group of a field implies characteristic in holomorph 0 en Additive group of a field implies characteristic in holomorph Class two implies generated by abelian normal subgroups 0 en Class two implies generated by abelian normal subgroups Abelian implies every subgroup is normal 0 en Abelian implies every subgroup is normal Finite supersolvable implies subgroups of all orders dividing the group order 0 en Finite supersolvable implies subgroups of all orders dividing the group order Odd-order cyclic group is characteristic in holomorph 0 en Odd-order cyclic group is characteristic in holomorph Finitely generated inner automorphism group implies every locally inner automorphism is inner 0 en Finitely generated inner automorphism group implies every locally inner automorphism is inner K implies Frattini-free 0 en K implies Frattini-free FZ implies finite derived subgroup 0 en FZ implies finite derived subgroup 4-Engel implies locally nilpotent for groups 0 en 4-Engel implies locally nilpotent for groups Normalizer condition implies locally nilpotent 0 en Normalizer condition implies locally nilpotent Noetherian implies Hopfian 0 en Noetherian implies Hopfian Finitely generated implies countable 0 en Finitely generated implies countable Nilpotent automorphism group implies nilpotent of class at most one more 0 en Nilpotent automorphism group implies nilpotent of class at most one more Characteristically simple implies CSCFN-realizable 0 en Characteristically simple implies CSCFN-realizable Solvable automorphism group implies solvable of derived length at most one more 0 en Solvable automorphism group implies solvable of derived length at most one more 2-Engel implies class three for groups 0 en 2-Engel implies class three for groups Finitely presented implies all homomorphisms to any finite group can be listed in finite time 0 en Finitely presented implies all homomorphisms to any finite group can be listed in finite time Finitely presented and residually finite implies solvable word problem 0 en Finitely presented and residually finite implies solvable word problem Schur multiplier of free group is trivial 0 en Schur multiplier of free group is trivial Finitely generated and free implies Hopfian 0 en Finitely generated and free implies Hopfian Abelian automorphism group implies class two 0 en Abelian automorphism group implies class two 3-Engel implies locally nilpotent for groups 0 en 3-Engel implies locally nilpotent for groups Primitive implies innately transitive 0 en Primitive implies innately transitive Abelian implies self-centralizing in holomorph 0 en Abelian implies self-centralizing in holomorph 3-step group implies solvable CN-group 0 en 3-step group implies solvable CN-group Conjugacy-separable implies every quotient-pullbackable automorphism is class-preserving 0 en Conjugacy-separable implies every quotient-pullbackable automorphism is class-preserving Cyclic implies abelian automorphism group 0 en Cyclic implies abelian automorphism group Centerless and maximal in automorphism group implies every automorphism is normal-extensible 0 en Centerless and maximal in automorphism group implies every automorphism is normal-extensible Conjugacy-separable and aperiodic implies every extensible automorphism is class-preserving 0 en Conjugacy-separable and aperiodic implies every extensible automorphism is class-preserving Finitely presented and conjugacy-separable implies solvable conjugacy problem 0 en Finitely presented and conjugacy-separable implies solvable conjugacy problem Proper and normal in quasisimple implies central 0 en Proper and normal in quasisimple implies central Abelian implies nilpotent 0 en Abelian implies nilpotent Dedekind implies class two 0 en Dedekind implies class two Finite abelian implies same orbit sizes of conjugacy classes and irreducible representations under automorphism group 0 en Finite abelian implies same orbit sizes of conjugacy classes and irreducible representations under automorphism group Free implies every subgroup is descendant 0 en Free implies every subgroup is descendant ACIC implies nilpotent (finite groups) 0 en ACIC implies nilpotent (finite groups) Finitely generated abelian implies Hopfian 0 en Finitely generated abelian implies Hopfian Normal implies modular 0 en Normal implies modular Locally finite Artinian p-group implies hypercentral 0 en Locally finite Artinian p-group implies hypercentral Identity functor controls strong fusion for saturated fusion system on abelian group 0 en Identity functor controls strong fusion for saturated fusion system on abelian group Algebraically closed implies simple 0 en Algebraically closed implies simple Finitely generated and nilpotent implies Hopfian 0 en Finitely generated and nilpotent implies Hopfian Finitely generated abelian implies residually finite 0 en Finitely generated abelian implies residually finite Noetherian implies subnormal join property 0 en Noetherian implies subnormal join property Supersolvable implies every nontrivial normal subgroup contains a cyclic normal subgroup 0 en Supersolvable implies every nontrivial normal subgroup contains a cyclic normal subgroup 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 Dedekind implies ACIC 0 en Dedekind implies ACIC Prime power order implies nilpotent 0 en Prime power order implies nilpotent Nilpotent implies normalizer condition 0 en Nilpotent implies normalizer condition Cyclic over central implies abelian 0 en Cyclic over central implies abelian Every finite group has a finite composition series 0 en Every finite group has a finite composition series P-Frattini-realizable implies not non-abelian cyclic-center 0 en P-Frattini-realizable implies not non-abelian cyclic-center Order has only two prime factors implies solvable 0 en Order has only two prime factors implies solvable Dihedral implies solvable 0 en Dihedral implies solvable Monomial implies solvable 0 en Monomial implies solvable Supersolvable implies nilpotent derived subgroup 0 en Supersolvable implies nilpotent derived subgroup Abelian implies every element is automorphic to its inverse 0 en Abelian implies every element is automorphic to its inverse Finite minimal simple implies 2-generated 0 en Finite minimal simple implies 2-generated Odd-order implies solvable 0 en Odd-order implies solvable Finitely generated and residually finite implies Hopfian 0 en Finitely generated and residually finite implies Hopfian Finitely generated implies finitely many homomorphisms to any finite group 0 en Finitely generated implies finitely many homomorphisms to any finite group Linearly primitive implies cyclic-center 0 en Linearly primitive implies cyclic-center Ascending chain condition on normal subgroups implies Hopfian 0 en Ascending chain condition on normal subgroups implies Hopfian Ascending chain condition on subnormal subgroups implies subnormal join property 0 en Ascending chain condition on subnormal subgroups implies subnormal join property Frattini-embedded normal-realizable implies every automorph-conjugate subgroup is characteristic 0 en Frattini-embedded normal-realizable implies every automorph-conjugate subgroup is characteristic Strongly p-solvable implies Glauberman type for odd p 0 en Strongly p-solvable implies Glauberman type for odd p P-solvable implies p-constrained 0 en P-solvable implies p-constrained Maximal among abelian normal implies self-centralizing in nilpotent 0 en Maximal among abelian normal implies self-centralizing in nilpotent Finitely many homomorphisms to any finite group implies every subgroup of finite index has finitely many automorphic subgroups 0 en Finitely many homomorphisms to any finite group implies every subgroup of finite index has finitely many automorphic subgroups Finite implies subnormal join property 0 en Finite implies subnormal join property Automorphism group is transitive on non-identity elements implies characteristically simple 0 en Automorphism group is transitive on non-identity elements implies characteristically simple Finite simple implies 2-generated 0 en Finite simple implies 2-generated Glauberman type implies ZJ-functor controls fusion 0 en Glauberman type implies ZJ-functor controls fusion Finite normal implies amalgam-characteristic 0 en Finite normal implies amalgam-characteristic Conjugacy-separable implies residually finite 0 en Conjugacy-separable implies residually finite Cyclic quotient of automorphism group by class-preserving automorphism group implies same orbit sizes of conjugacy classes and irreducible representations under automorphism group 0 en Cyclic quotient of automorphism group by class-preserving automorphism group implies same orbit sizes of conjugacy classes and irreducible representations under automorphism group Nilpotent implies solvable 0 en Nilpotent implies solvable Nilpotent implies every subgroup is subnormal 0 en Nilpotent implies every subgroup is subnormal Every subgroup is subnormal implies normalizer condition 0 en Every subgroup is subnormal implies normalizer condition Schur multiplier of cyclic group is trivial 0 en Schur multiplier of cyclic group is trivial Class-inverting automorphism implies every element is automorphic to its inverse 0 en Class-inverting automorphism implies every element is automorphic to its inverse Cyclic implies abelian 0 en Cyclic implies abelian Nilpotent derived subgroup implies subnormal join property 0 en Nilpotent derived subgroup implies subnormal join property Nilpotent implies every nontrivial normal subgroup contains a cyclic normal subgroup 0 en Nilpotent implies every nontrivial normal subgroup contains a cyclic normal subgroup Nilpotent implies center is normality-large 0 en Nilpotent implies center is normality-large Center is normality-large implies every nontrivial normal subgroup contains a cyclic normal subgroup 0 en Center is normality-large implies every nontrivial normal subgroup contains a cyclic normal subgroup Metabelian implies subnormal join property 0 en Metabelian implies subnormal join property Finitely generated abelian is subgroup-closed 0 en Finitely generated abelian is subgroup-closed Frattini-embedded normal-realizable implies inner-in-automorphism-Frattini 0 en Frattini-embedded normal-realizable implies inner-in-automorphism-Frattini Extraspecial implies Camina 0 en Extraspecial implies Camina Prime power order implies subgroups of all orders dividing the group order 0 en Prime power order implies subgroups of all orders dividing the group order Schur multiplier of Z-group is trivial 0 en Schur multiplier of Z-group is trivial Nilpotent implies every maximal subgroup is normal 0 en Nilpotent implies every maximal subgroup is normal Normalizer condition implies every maximal subgroup is normal 0 en Normalizer condition implies every maximal subgroup is normal Multiplicative group of a field implies every finite subgroup is cyclic 0 en Multiplicative group of a field implies every finite subgroup is cyclic At most n elements of order dividing n implies every finite subgroup is cyclic 0 en At most n elements of order dividing n implies every finite subgroup is cyclic Free implies residually nilpotent 0 en Free implies residually nilpotent FZ implies generalized subnormal join property 0 en FZ implies generalized subnormal join property Abelian implies ACIC 0 en Abelian implies ACIC Locally cyclic implies abelian automorphism group 0 en Locally cyclic implies abelian automorphism group Artinian implies co-Hopfian 0 en Artinian implies co-Hopfian Algebra group implies power degree group for field size 0 en Algebra group implies power degree group for field size Free implies residually finite 0 en Free implies residually finite SQ-universal implies no nontrivial identity 0 en SQ-universal implies no nontrivial identity Additive group of a field implies characteristic in holomorph 0 en Additive group of a field implies characteristic in holomorph Class two implies generated by abelian normal subgroups 0 en Class two implies generated by abelian normal subgroups Abelian implies every subgroup is normal 0 en Abelian implies every subgroup is normal Finite supersolvable implies subgroups of all orders dividing the group order 0 en Finite supersolvable implies subgroups of all orders dividing the group order Odd-order cyclic group is characteristic in holomorph 0 en Odd-order cyclic group is characteristic in holomorph Finitely generated inner automorphism group implies every locally inner automorphism is inner 0 en Finitely generated inner automorphism group implies every locally inner automorphism is inner K implies Frattini-free 0 en K implies Frattini-free FZ implies finite derived subgroup 0 en FZ implies finite derived subgroup 4-Engel implies locally nilpotent for groups 0 en 4-Engel implies locally nilpotent for groups Normalizer condition implies locally nilpotent 0 en Normalizer condition implies locally nilpotent Noetherian implies Hopfian 0 en Noetherian implies Hopfian Finitely generated implies countable 0 en Finitely generated implies countable Nilpotent automorphism group implies nilpotent of class at most one more 0 en Nilpotent automorphism group implies nilpotent of class at most one more Characteristically simple implies CSCFN-realizable 0 en Characteristically simple implies CSCFN-realizable Solvable automorphism group implies solvable of derived length at most one more 0 en Solvable automorphism group implies solvable of derived length at most one more 2-Engel implies class three for groups 0 en 2-Engel implies class three for groups Finitely presented implies all homomorphisms to any finite group can be listed in finite time 0 en Finitely presented implies all homomorphisms to any finite group can be listed in finite time Finitely presented and residually finite implies solvable word problem 0 en Finitely presented and residually finite implies solvable word problem Schur multiplier of free group is trivial 0 en Schur multiplier of free group is trivial Finitely generated and free implies Hopfian 0 en Finitely generated and free implies Hopfian Abelian automorphism group implies class two 0 en Abelian automorphism group implies class two 3-Engel implies locally nilpotent for groups 0 en 3-Engel implies locally nilpotent for groups Primitive implies innately transitive 0 en Primitive implies innately transitive Abelian implies self-centralizing in holomorph 0 en Abelian implies self-centralizing in holomorph 3-step group implies solvable CN-group 0 en 3-step group implies solvable CN-group Conjugacy-separable implies every quotient-pullbackable automorphism is class-preserving 0 en Conjugacy-separable implies every quotient-pullbackable automorphism is class-preserving Cyclic implies abelian automorphism group 0 en Cyclic implies abelian automorphism group Centerless and maximal in automorphism group implies every automorphism is normal-extensible 0 en Centerless and maximal in automorphism group implies every automorphism is normal-extensible Conjugacy-separable and aperiodic implies every extensible automorphism is class-preserving 0 en Conjugacy-separable and aperiodic implies every extensible automorphism is class-preserving Finitely presented and conjugacy-separable implies solvable conjugacy problem 0 en Finitely presented and conjugacy-separable implies solvable conjugacy problem Proper and normal in quasisimple implies central 0 en Proper and normal in quasisimple implies central Abelian implies nilpotent 0 en Abelian implies nilpotent Dedekind implies class two 0 en Dedekind implies class two Finite abelian implies same orbit sizes of conjugacy classes and irreducible representations under automorphism group 0 en Finite abelian implies same orbit sizes of conjugacy classes and irreducible representations under automorphism group Free implies every subgroup is descendant 0 en Free implies every subgroup is descendant ACIC implies nilpotent (finite groups) 0 en ACIC implies nilpotent (finite groups) Finitely generated abelian implies Hopfian 0 en Finitely generated abelian implies Hopfian Normal implies modular 0 en Normal implies modular Locally finite Artinian p-group implies hypercentral 0 en Locally finite Artinian p-group implies hypercentral Identity functor controls strong fusion for saturated fusion system on abelian group 0 en Identity functor controls strong fusion for saturated fusion system on abelian group Algebraically closed implies simple 0 en Algebraically closed implies simple Finitely generated and nilpotent implies Hopfian 0 en Finitely generated and nilpotent implies Hopfian INST 132 en INST