# Semantic search

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