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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