Cyclic automorphism group not implies cyclic
Glauberman type not implies p-constrained
Locally finite not implies embeddable in finitary symmetric group
Abelian automorphism group not implies abelian
Finitely generated and parafree not implies free
Isomorphic to inner automorphism group not implies centerless
Characteristically metacyclic not implies metacyclic derived series
P-constrained not implies Glauberman type for p
Subgroups of all orders dividing the group order not implies supersolvable
Nilpotent not implies nilpotent automorphism group
Torsion-free not implies powering-injective
Noetherian not implies finitely presented
Finitely generated not implies Noetherian
Golod's theorem on locally finite groups
Finitely presented and solvable not implies polycyclic
Ambivalent not implies strongly ambivalent
Solvable and generated by finitely many periodic elements not implies periodic
Finitely generated and solvable not implies finitely presented
Finitely generated not implies finitely presented
Nilpotent automorphism group not implies abelian automorphism group
2-Engel not implies class two for groups
Finitely presented not implies Noetherian
Sylow tower not implies subgroups of all orders dividing the group order
P-solvable not implies Glauberman type for p
Rational not implies rational-representation
There exist abelian groups whose isomorphism classes of direct powers have any given period
Subgroups of all orders dividing the group order not implies Sylow tower
Same Hall-Senior genus not implies character table-equivalent
Finite solvable not implies supersolvable
Locally nilpotent not implies normalizer condition
Solvable not implies solvable automorphism group
Finite solvable not implies subgroups of all orders dividing the group order
Class two not implies abelian automorphism group
Finitely generated and solvable not implies polycyclic
Abelian automorphism group not implies cyclic
Rationally powered not implies nilpotent
Finite not implies composition factor-unique
Composition factor-unique not implies composition series-unique
No proper nontrivial transitively normal subgroup not implies simple
Residually finite not implies finitely generated
Residually nilpotent not implies imperfect
Divisible not implies rationally powered
Nilpotent not implies generated by abelian normal subgroups
Finitely generated not implies residually finite
Normalizer condition not implies nilpotent
Finite not implies composition factor-permutable