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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