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1-closed transversal not implies permutably complemented, 2-Engel implies class three for groups, 2-Engel not implies class two for groups, 2-Sylow subgroup is TI implies it is normal or there is exactly one conjugacy class of involutions, 2-hypernormalized satisfies intermediate subgroup condition, 2-subnormal implies conjugate-join-closed subnormal, 2-subnormal implies conjugate-permutable, 2-subnormal implies join-transitively subnormal, 2-subnormal not implies automorph-permutable, 2-subnormal not implies hypernormalized, 2-subnormality is conjugate-join-closed, 2-subnormality is not finite-join-closed, 2-subnormality is not finite-upper join-closed, 2-subnormality is not transitive, 2-subnormality is strongly intersection-closed, 3-Engel implies locally nilpotent for groups, 3-step group implies solvable CN-group, 3-subnormal implies finite-conjugate-join-closed subnormal, 3-subnormal not implies finite-automorph-join-closed subnormal, 4-Engel implies locally nilpotent for groups, 4-subnormal not implies finite-conjugate-join-closed subnormal, ACIC implies nilpotent (finite groups), ACIC is characteristic subgroup-closed, AEP does not satisfy intermediate subgroup condition, Abelian and pronormal implies SCDIN, Abelian automorphism group implies class two, Abelian automorphism group not implies abelian, Abelian automorphism group not implies cyclic, Abelian characteristic is not join-closed, Abelian implies ACIC, Abelian implies every element is automorphic to its inverse, Abelian implies every subgroup is normal, Abelian implies nilpotent, Abelian implies self-centralizing in holomorph, Abelian normal is not join-closed, Abelian normal not implies central, Abelian normal subgroup of core-free maximal subgroup is contranormal implies derived subgroup of whole group is monolith, Abelian p-group with indecomposable coprime automorphism group is homocyclic, Abelian-quotient not implies cocentral, Abelian-quotient not implies kernel of a bihomomorphism, Abelianness is 2-local, Abelianness is directed union-closed, Abelianness is quotient-closed, Abelianness is subgroup-closed, Abnormal implies WNSCC, Abnormal normalizer not implies pronormal, Additive group of a field implies characteristic in holomorph, Algebra group implies power degree group for field size, Algebraically closed implies simple, All cumulative conjugacy class size statistics values divide the order of the group for groups up to prime-fifth order, All partial sum values of squares of degrees of irreducible representations divide the order of the group for groups up to prime-fifth order, Alperin's fusion theorem in terms of well-placed tame intersections, Alternating implies flexible, Alternative implies powers up to the fifth are well-defined, Amalgam-characteristic implies image-potentially characteristic, Amalgam-characteristic implies potentially characteristic, Ambivalence is direct product-closed, Ambivalence is quotient-closed, Ambivalent not implies strongly ambivalent, Analogue of Thompson transitivity theorem fails for abelian subgroups of rank two, Analogue of Thompson transitivity theorem fails for groups in which not every p-local subgroup is p-constrained, Any abelian normal subgroup normalizes an abelian subgroup of maximum order, Any class two normal subgroup whose derived subgroup is in the ZJ-subgroup normalizes an abelian subgroup of maximum order, Artinian implies co-Hopfian, Artinian implies periodic, Ascendant not implies subnormal, Ascending chain condition on normal subgroups implies Hopfian, Ascending chain condition on subnormal subgroups implies subnormal join property, Ascending chain condition on subnormal subgroups is normal subgroup-closed, Associative implies generalized associative, At most n elements of order dividing n implies every finite subgroup is cyclic, Automorph-conjugacy is centralizer-closed, Automorph-conjugacy is normalizer-closed, Automorph-conjugacy is not finite-conjugate-intersection-closed, Automorph-conjugacy is not finite-intersection-closed, Automorph-conjugacy is not finite-join-closed, Automorph-conjugacy is transitive, Automorph-permutable not implies permutable, Automorphism group is transitive on non-identity elements implies characteristically simple, Baer Lie property is not quotient-closed, Baer Lie property is not subgroup-closed, Base of a wreath product implies right-transitively 2-subnormal, Base of a wreath product implies right-transitively conjugate-permutable, Base of a wreath product implies subset-conjugacy-closed, Base of a wreath product is transitive, Base of a wreath product not implies elliptic, Brauer's induction theorem, Brauer-Fowler inequality relating number of conjugacy classes of strongly real elements and number of involutions, Brauer-Fowler theorem on existence of subgroup of order greater than the cube root of the group order, Bryant-Kovacs theorem, Burnside's basis theorem, Burnside's theorem on coprime automorphisms and Frattini subgroup, C-closed implies local powering-invariant, C-closed implies powering-invariant, CA not implies nilpotent, CDIN of conjugacy-closed implies CDIN, CEP implies every relatively normal subgroup is weakly closed, Cayley's theorem, Center is normality-large implies every nontrivial normal subgroup contains a cyclic normal subgroup, Center of pronormal implies SCDIN, Center-fixing implies central factor-extensible, Centerless and maximal in automorphism group implies every automorphism is normal-extensible, Central factor implies normal, Central factor implies transitively normal, Central factor is centralizer-closed, Central factor is not finite-intersection-closed, Central factor is not finite-join-closed, Central factor is not quotient-transitive, Central factor is transitive, Central factor is upper join-closed, Central factor not implies direct factor, Central factor satisfies image condition, Central factor satisfies intermediate subgroup condition, Central implies abelian normal, Central implies amalgam-characteristic, Central implies image-potentially characteristic, Central implies normal, Central implies normal satisfying the subgroup-to-quotient powering-invariance implication, Central implies potentially characteristic, Central implies potentially verbal in finite, Central product decomposition lemma for characteristic rank one, Central subgroup implies join-transitively central factor, Centralizer of coprime automorphism in homomorphic image equals image of centralizer, Centralizer product theorem, Centralizer product theorem for elementary abelian group, Centralizer-commutator product decomposition for finite groups and cyclic automorphism group, Centralizer-commutator product decomposition for finite nilpotent groups, Centralizer-free ideal implies automorphism-faithful, Centralizer-free ideal implies derivation-faithful, Characteristic Lie subring not implies ideal, Characteristic and self-centralizing implies coprime automorphism-faithful, Characteristic central factor of WNSCDIN implies WNSCDIN, Characteristic direct factor not implies fully invariant, Characteristic implies automorph-conjugate, Characteristic implies normal, Characteristic not implies amalgam-characteristic, Characteristic not implies characteristic-isomorph-free in finite, Characteristic not implies derivation-invariant, Characteristic not implies direct factor, Characteristic not implies elementarily characteristic, Characteristic not implies fully invariant, Characteristic not implies fully invariant in finite abelian group, Characteristic not implies fully invariant in finitely generated abelian group, Characteristic not implies fully invariant in odd-order class two p-group, Characteristic not implies injective endomorphism-invariant, Characteristic not implies injective endomorphism-invariant in finitely generated abelian group, Characteristic not implies isomorph-free in finite group, Characteristic not implies isomorph-normal in finite group, Characteristic not implies normal in loops, Characteristic not implies normal-isomorph-free, Characteristic not implies potentially fully invariant, Characteristic not implies powering-invariant in nilpotent group, Characteristic not implies powering-invariant in solvable group, Characteristic not implies quasiautomorphism-invariant, Characteristic not implies strictly characteristic, Characteristic not implies sub-(isomorph-normal characteristic) in finite, Characteristic not implies sub-isomorph-free in finite group, Characteristic of CDIN implies CDIN, Characteristic of normal implies normal, Characteristic rank one is characteristic subgroup-closed, Characteristic subgroup of Sylow subgroup is weakly closed iff it is normal in every Sylow subgroup containing it, Characteristic subgroup of abelian group implies intermediately powering-invariant, Characteristic subgroup of abelian group implies powering-invariant, Characteristic subgroup of abelian group is quotient-powering-invariant, Characteristic subgroup of abelian group not implies divisibility-closed, Characteristic subgroup of abelian group not implies local powering-invariant, Characteristic upper-hook AEP implies characteristic, Characteristic-isomorph-free not implies normal-isomorph-free in finite, Characteristically complemented characteristic is transitive, Characteristically metacyclic and commutator-realizable implies abelian, Characteristically metacyclic not implies metacyclic derived series, Characteristically simple implies CSCFN-realizable, Characteristicity does not satisfy image condition, Characteristicity does not satisfy intermediate subgroup condition, Characteristicity does not satisfy lower central series condition, Characteristicity is centralizer-closed, Characteristicity is commutator-closed, Characteristicity is not finite direct power-closed, Characteristicity is not finite-relative-intersection-closed, Characteristicity is not upper join-closed, Characteristicity is quotient-transitive, Characteristicity is strongly intersection-closed, Characteristicity is strongly join-closed, Characteristicity is transitive, Characteristicity is transitive for Lie rings, Characteristicity satisfies partition difference condition, Class two implies generated by abelian normal subgroups, Class two not implies abelian automorphism group, Class-inverting automorphism implies every element is automorphic to its inverse, Class-inverting automorphism induces class-inverting automorphism on any quotient, Class-preserving implies IA, Class-preserving implies linearly extensible, Class-preserving implies linearly pushforwardable, Class-preserving not implies inner, Class-preserving not implies subgroup-conjugating, Classification of extraspecial groups, Classification of finite 2-groups of maximal class, Classification of finite p-groups of characteristic rank one, Classification of finite p-groups of normal rank one, Classification of finite p-groups of rank one, Classification of finite p-groups with cyclic maximal subgroup, Classification of finite p-groups with cyclic normal self-centralizing subgroup, Classification of finite solvable CN-groups, Clifford's theorem, Cocentral implies central factor, Cocentral implies centralizer-dense, Cocentral implies right-quotient-transitively central factor, Cocentral not implies amalgam-characteristic, Cocentrality is transitive, Cocentrality is upward-closed, Cocentrality satisfies intermediate subgroup condition, Cofactorial automorphism-invariance is not transitive, Cofactorial automorphism-invariant implies left-transitively 2-subnormal, Column orthogonality theorem, Commensurator of subgroup is subgroup, Commutative implies flexible, Commutator of a group and a subgroup implies normal, Commutator of a normal subgroup and a subset implies 2-subnormal, Commutator of a transitively normal subgroup and a subset implies normal, Commutator of finite group with cyclic coprime automorphism group equals second commutator, Commutator of finite nilpotent group with coprime automorphism group equals second commutator, Commutator-in-center is intersection-closed, Commuting of non-identity elements defines an equivalence relation between prime divisors of the order of a finite CN-group, Complemented central factor not implies direct factor, Complemented characteristic not implies left-transitively complemented normal, Complemented normal implies endomorphism kernel, Complemented normal is quotient-transitive, Complemented normal not implies direct factor, Complemented normal not implies local powering-invariant, Complete divisibility-closedness is strongly intersection-closed, Complete divisibility-closedness is transitive, Complete not implies ambivalent, Composition factor-unique not implies composition series-unique, Conjugacy class of prime power size implies not simple, Conjugacy functor whose normalizer generates whole group with p'-core controls fusion, Conjugacy-closed and Hall not implies retract, Conjugacy-closed implies focal subgroup equals derived subgroup, Conjugacy-closed normal not implies central factor, Conjugacy-closed not implies weak subset-conjugacy-closed, Conjugacy-closedness is not join-closed, Conjugacy-closedness is not upper join-closed, Conjugacy-closedness is transitive, Conjugacy-separable and aperiodic implies every extensible automorphism is class-preserving, Conjugacy-separable implies every quotient-pullbackable automorphism is class-preserving, Conjugacy-separable implies residually finite, Conjugate-comparable not implies normal, Conjugate-denseness is transitive, Conjugate-intersection index theorem, Conjugate-join-closed subnormal implies join-transitively subnormal, Conjugate-permutability is conjugate-join-closed, Conjugate-permutability satisfies intermediate subgroup condition, Contranormal not implies self-normalizing, Contranormality is UL-join-closed, Contranormality is transitive, Contranormality is upper join-closed, Coprime automorphism-faithful characteristicity is transitive, Core-free and permutable implies subdirect product of finite nilpotent groups, Core-free permutable subnormal implies solvable of length at most one less than subnormal depth, Corollary of Thompson transitivity theorem, Corollary of centralizer product theorem for rank at least three, Criterion for projective representation to lift to linear representation, Cube map is endomorphism iff abelian (if order is not a multiple of 3), Cube map is surjective endomorphism implies abelian, Cyclic Frattini quotient implies cyclic, Cyclic automorphism group not implies cyclic, Cyclic characteristic implies hereditarily characteristic, Cyclic iff not a union of proper subgroups, Cyclic implies abelian, Cyclic implies abelian automorphism group, Cyclic normal implies hereditarily normal, Cyclic normal is not join-closed, Cyclic over central implies abelian, Cyclic quotient of automorphism group by class-preserving automorphism group implies same orbit sizes of conjugacy classes and irreducible representations under automorphism group, Cyclic-quotient characteristic implies upward-closed characteristic, Cyclicity is subgroup-closed, Dedekind implies ACIC, Dedekind implies class two, Dedekind not implies abelian, Derivation-invariance does not satisfy intermediate subring condition, Derivation-invariance is Lie bracket-closed, Derivation-invariance is centralizer-closed, Derivation-invariance is not upper join-closed, Derivation-invariance is transitive, Derivation-invariant not implies characteristic, Derived subgroup satisfies ascending chain condition on subnormal subgroups implies subnormal join property, Descendance does not satisfy image condition, Descendant not implies subnormal, Dickson's theorem, Dihedral trick, Direct factor implies central factor, Direct factor implies join-transitively central factor, Direct factor implies normal, Direct factor implies right-quotient-transitively central factor, Direct factor implies transitively normal, Direct factor is not finite-join-closed, Direct factor is not intersection-closed, Direct factor is not upper join-closed, Direct factor is quotient-transitive, Direct factor is transitive, Direct factor not implies amalgam-characteristic, Direct factor not implies characteristic, Direct factor satisfies intermediate subgroup condition, Direct factor satisfies lower central series condition, Directed union of subgroups is subgroup, Divisibility is central extension-closed, Divisibility-closed implies powering-invariant, Divisibility-closed not implies local divisibility-closed, Divisibility-closedness is not finite-intersection-closed, Divisibility-closedness is not finite-join-closed, Divisibility-closedness is transitive, Divisible not implies rationally powered, Double coset index two implies maximal, Doubly transitive implies primitive, Endomorph-dominating not implies automorph-conjugate, Endomorphism image implies divisibility-closed, Endomorphism image implies powering-invariant, Endomorphism kernel does not satisfy intermediate subgroup condition, Endomorphism kernel implies quotient-powering-invariant, Endomorphism kernel is not transitive, Endomorphism kernel is quotient-transitive, Endomorphism kernel not implies complemented normal, Equivalence of definitions of finite characteristically simple group, Equivalence of definitions of group, Every Sylow subgroup is cyclic implies metacyclic, Every element is automorphic to its inverse is characteristic subgroup-closed, Every finite division ring is a field, Every group is a quotient of a free group, Every group is a quotient of a hypoabelian group, Every group is a quotient of a residually finite group, Every group is a quotient of a residually nilpotent group, Every group is a union of cyclic subgroups, Every group is characteristic in itself, Every group is normal in itself, Every nontrivial subgroup of the group of integers is cyclic on its smallest element, Every normal subgroup satisfies the quotient-to-subgroup powering-invariance implication, Every subgroup is subnormal implies normalizer condition, Exactly n elements of order dividing n in a finite solvable group implies the elements form a subgroup, Existentially bound-word not implies verbal, Extensible implies normal, Extensible implies permutation-extensible, Extensible implies subgroup-conjugating, Extraspecial commutator-in-center subgroup is central factor, Extraspecial implies Camina, FC not implies BFC, FC not implies finite derived subgroup, FZ implies finite derived subgroup, FZ implies generalized subnormal join property, Finite N-group is solvable or almost simple, Finite abelian implies same orbit sizes of conjugacy classes and irreducible representations under automorphism group, Finite derived subgroup not implies FZ, Finite direct power-closed characteristic is quotient-transitive, Finite direct power-closed characteristic is transitive, Finite direct power-closed characteristic not implies fully invariant, Finite double coset index is not finite-intersection-closed, Finite group having at least two conjugacy classes of involutions has order less than the cube of the maximum of orders of centralizers of involutions, Finite implies local powering-invariant, Finite implies powering-invariant, Finite implies subnormal join property, Finite index implies completely divisibility-closed, Finite index implies powering-invariant, Finite index not implies local powering-invariant, Finite minimal simple implies 2-generated, Finite non-abelian 2-group has maximal class iff its abelianization has order four, Finite normal implies amalgam-characteristic, Finite normal implies image-potentially characteristic, Finite normal implies quotient-powering-invariant, Finite not implies composition factor-permutable, Finite not implies composition factor-unique, Finite not implies divisibility-closed in abelian group, Finite simple implies 2-generated, Finite solvable not implies p-normal, Finite solvable not implies subgroups of all orders dividing the group order, Finite solvable not implies supersolvable, Finite solvable-extensible implies class-preserving, Finite solvable-extensible implies inner, Finite supersolvable implies subgroups of all orders dividing the group order, Finite-extensible implies Hall-semidirectly extensible, Finite-extensible implies class-preserving, Finite-extensible implies inner, Finite-extensible implies subgroup-conjugating, Finite-quotient-pullbackable implies class-preserving, Finite-quotient-pullbackable implies inner, Finitely generated abelian implies Hopfian, Finitely generated abelian implies residually finite, Finitely generated abelian is subgroup-closed, Finitely generated and free implies Hopfian, Finitely generated and nilpotent implies Hopfian, Finitely generated and parafree not implies free, Finitely generated and residually finite implies Hopfian, Finitely generated and solvable not implies finitely presented, Finitely generated and solvable not implies polycyclic, Finitely generated implies countable, Finitely generated implies every subgroup of finite index has finitely many automorphic subgroups, Finitely generated implies finitely many homomorphisms to any finite group, Finitely generated inner automorphism group implies every locally inner automorphism is inner, Finitely generated not implies Noetherian, Finitely generated not implies finitely presented, Finitely generated not implies residually finite, Finitely many homomorphisms to any finite group implies every subgroup of finite index has finitely many automorphic subgroups, Finitely presented and conjugacy-separable implies solvable conjugacy problem, Finitely presented and residually finite implies solvable word problem, Finitely presented and solvable not implies polycyclic, Finitely presented implies all homomorphisms to any finite group can be listed in finite time, Finitely presented not implies Noetherian, Finiteness is extension-closed, First isomorphism theorem, Fixed-point subgroup of a subgroup of the automorphism group implies local powering-invariant, Fixed-point-free automorphism of order four implies solvable, Fixed-point-free automorphism of order three implies nilpotent, Fixed-point-free involution on finite group is inverse map, Focal subgroup of a Sylow subgroup is generated by the commutators with normalizers of non-identity tame intersections, Focal subgroup theorem, Fourth isomorphism theorem, Frattini subgroup is nilpotent in finite, Frattini subgroup is normal-monotone, Frattini's argument, Frattini-embedded normal-realizable implies every automorph-conjugate subgroup is characteristic, Frattini-embedded normal-realizable implies inner-in-automorphism-Frattini, Frattini-in-center odd-order p-group implies p-power map is endomorphism, Free abelian is subgroup-closed, Free factor implies self-normalizing or trivial, Free implies every subgroup is descendant, Free implies residually finite, Free implies residually nilpotent, Freeness is subgroup-closed, Frobenius' normal p-complement theorem, Full invariance does not satisfy image condition, Full invariance does not satisfy intermediate subgroup condition, Full invariance is finite direct power-closed, Full invariance is not direct power-closed, Full invariance is quotient-transitive, Full invariance is strongly join-closed, Full invariance is transitive, Fully invariant direct factor implies left-transitively homomorph-containing, Fully invariant implies characteristic, Fully invariant implies finite direct power-closed characteristic, Fully invariant implies ideal for class two Lie ring, Fully invariant not implies abelian-potentially verbal in abelian group, Fully invariant not implies normal in loops, Fully invariant not implies verbal in finite abelian group, Fully invariant of strictly characteristic implies strictly characteristic, Fully invariant subgroup of abelian group not implies divisibility-closed, Fusion system-relatively weakly closed is not finite-intersection-closed, Fusion system-relatively weakly closed not implies isomorph-normal, Glauberman type implies ZJ-functor controls fusion, Glauberman type is not quotient-closed, Glauberman type not implies p-constrained, Glauberman's replacement theorem, Glauberman's theorem on intersection with the ZJ-subgroup, Glauberman-Thompson normal p-complement theorem, Golod's theorem on locally finite groups, Grand orthogonality theorem, Group acts as automorphisms by conjugation, Group implies G-loop, Grün's first theorem on the focal subgroup, Hall does not satisfy transfer condition, Hall implies join of Sylow subgroups, Hall implies paracharacteristic, Hall is transitive, Hall not implies WNSCDIN, Hall not implies automorph-conjugate, Hall not implies order-conjugate, Hall not implies order-isomorphic, Hall not implies procharacteristic, Hall not implies pronormal, Hall retract implies order-conjugate, Hall satisfies intermediate subgroup condition, Hall satisfies permuting transfer condition, Hall subgroups exist in finite solvable group, Hall subgroups need not exist, Hall-extensible implies class-preserving, Hall-semidirectly extensible implies inner, Hall-semidirectly extensible implies linearly pushforwardable over prime field, Having subgroups of all orders dividing the group order is not quotient-closed, Having subgroups of all orders dividing the group order is not subgroup-closed, Hereditarily characteristic not implies cyclic in finite, Higman's theorem on automorphism of prime order of Lie ring, Homomorph-containing not implies no nontrivial homomorphism to quotient group, Homomorph-containment is finite direct power-closed, Homomorph-containment is not transitive, Homomorph-containment is quotient-transitive, Homomorph-containment is strongly join-closed, Homomorph-containment satisfies intermediate subgroup condition, Hopfianness is not subgroup-closed, Hypoabelian not implies imperfect, IA not implies class-preserving, Ideal not implies derivation-invariant, Ideal property is centralizer-closed, Ideal property is not transitive for Lie rings, Ideal property is upper join-closed for Lie rings, Identity functor controls strong fusion for abelian Sylow subgroup, Identity functor controls strong fusion for saturated fusion system on abelian group, Image-closed characteristic not implies fully invariant, Image-closed fully invariant not implies verbal, Imperfect not implies hypoabelian, Index four implies 2-subnormal or double coset index two, Index is multiplicative, Index three implies normal or double coset index two, Index two not implies characteristic, Inner implies IA