2-Sylow subgroup of rational group is rational if its class is at most two | | Finite group (?) Rational group (?) Group of nilpotency class two (?) |

Abelian direct factor implies potentially verbal in finite | | Finite group (?) Abelian direct factor (?) Potentially verbal subgroup (?) Abelian direct factor of finite group (2) Potentially verbal subgroup of finite group (3) |

Central implies finite-pi-potentially verbal in finite | | Finite group (?) Central subgroup (?) Finite-pi-potentally verbal subgroup (?) Central subgroup of finite group (2) Finite-pi-potentally verbal subgroup of finite group (3) Verbal subgroup (?) |

Central implies potentially fully invariant in finite | | Finite group (?) Central subgroup (?) Potentially fully invariant subgroup (?) Central subgroup of finite group (2) Potentially fully invariant subgroup of finite group (3) Finite-potentially fully invariant subgroup (?) Finite-potentially fully invariant subgroup (3) |

Central implies potentially verbal in finite | | Central subgroup of finite group (2) Finite-potentially verbal subgroup (2) Finite group (?) Central subgroup (?) Potentially verbal subgroup (?) Potentially verbal subgroup of finite group (3) Verbal subgroup (?) |

Characteristic not implies isomorph-free in finite group | | Finite group (2) Characteristic subgroup (2) Isomorph-free subgroup (2) |

Characteristic not implies isomorph-normal in finite group | | Finite group (2) Characteristic subgroup (2) Isomorph-normal subgroup (2) |

Characteristic not implies normal-isomorph-free | | Finite group (2) Characteristic subgroup (2) Normal-isomorph-free subgroup (2) |

Characteristic not implies sub-(isomorph-normal characteristic) in finite | | Finite group (2) Characteristic subgroup (2) Sub-(isomorph-normal characteristic) subgroup (2) |

Characteristic not implies sub-isomorph-free in finite group | | Finite group (2) Characteristic subgroup (2) Sub-isomorph-free subgroup (2) |

Conjecture that most finite groups are nilpotent | | Finite group (?) Order of a group (?) Finite nilpotent group (?) |

Conjugate-permutable implies subnormal in finite | | Finite group (?) Conjugate-permutable subgroup (?) Subnormal subgroup (?) Conjugate-permutable subgroup of finite group (2) Subnormal subgroup of finite group (3) |

Cyclic normal implies finite-pi-potentially verbal in finite | | Finite group (?) Cyclic normal subgroup (?) Finite-pi-potentially verbal subgroup (?) Cyclic normal subgroup of finite group (2) Finite-pi-potentially verbal subgroup of finite group (3) |

Cyclic normal implies potentially verbal in finite | | Finite group (?) Cyclic normal subgroup (?) Potentially verbal subgroup (?) Cyclic normal subgroup of finite group (2) Potentially verbal subgroup of finite group (3) |

Finite group implies cyclic iff every subgroup is characteristic | | Finite cyclic group (1) Finite group (?) Cyclic group (?) Characteristic subgroup (?) Characteristic subgroup of finite group (?) Fully invariant subgroup (?) Fully invariant subgroup of finite group (?) Isomorph-free subgroup (?) Isomorph-free subgroup of finite group (?) Verbal subgroup (?) Verbal subgroup of finite group (?) Homomorph-containing subgroup (?) Homomorph-containing subgroup of finite group (?) |

Finite groups are 1-isomorphic iff their directed power graphs are isomorphic | | 1-isomorphic finite groups (1) Finite group (?) 1-isomorphic groups (?) 1-isomorphic finite groups (?) Directed power graph of a group (?) |

Finite implies subnormal join property | | Finite group (2) Group satisfying subnormal join property (2) |

Finite normal implies amalgam-characteristic | | Finite normal subgroup (2) Amalgam-characteristic subgroup (2) Finite group (2) Always amalgam-characteristic group (2) |

Finite not implies composition factor-permutable | | Finite group (2) Composition factor-permutable group (2) |

Finite not implies composition factor-unique | | Finite group (2) Composition factor-unique group (2) |

Finitely many subgroups iff finite | | Finite group (?) Order of a group (?) |

Finiteness is extension-closed | | Finite group (1) Extension-closed group property (2) |

Fitting subgroup is normal-isomorph-free in finite | | Finite group (3) Fitting subgroup (2) Normal-isomorph-free subgroup (2) |

Frattini subgroup is nilpotent in finite | | Finite group (?) |

Hall not implies automorph-conjugate | | Finite group (2) Hall subgroup (2) Automorph-conjugate subgroup (2) |

Hall not implies order-isomorphic | | Finite group (2) Hall subgroup (2) Order-isomorphic subgroup (2) |

Hall not implies procharacteristic | | Finite group (2) Hall subgroup (2) Automorph-conjugate subgroup (2) |

Hereditarily characteristic not implies cyclic in finite | | Finite group (2) Hereditarily characteristic subgroup (2) Cyclic characteristic subgroup (2) Hereditarily characteristic subgroup (?) |

Homocyclic normal implies finite-pi-potentially fully invariant in finite | | Finite group (?) Homocyclic normal subgroup (?) Finite-pi-potentially fully invariant subgroup (?) Homocyclic normal subgroup of finite group (2) Finite-pi-potentially fully invariant subgroup of finite group (3) |

Homocyclic normal implies potentially fully invariant in finite | | Finite group (?) Homocyclic normal subgroup (?) Finite-potentially fully invariant subgroup (?) Homocyclic normal subgroup of finite group (2) Finite-potentially fully invariant subgroup (3) Potentially fully invariant subgroup (?) Potentially fully invariant subgroup of finite group (3) Fully invariant subgroup (?) |

Isomorph-conjugacy is normalizer-closed in finite | | Finite group (?) Isomorph-conjugate subgroup (?) Normalizer-closed subgroup property (?) |

Monolith is fully invariant in co-Hopfian group | | Co-Hopfian group (?) Finite group (?) Monolith (?) |

Nilpotent Hall implies isomorph-conjugate | | Finite group (?) Nilpotent Hall subgroup (?) Isomorph-conjugate subgroup (?) Nilpotent Hall subgroup of finite group (2) Isomorph-conjugate subgroup of finite group (3) Hall subgroup (?) Nilpotent group (?) |

Normal not implies finite-pi-potentially characteristic in finite | | Finite group (2) Normal subgroup (2) Finite-pi-potentially characteristic subgroup (2) |

Normal not implies normal-extensible automorphism-invariant in finite | | Finite group (2) Normal subgroup (2) Normal-extensible automorphism-invariant subgroup (2) |

Periodic not implies locally finite | | Periodic group (2) Locally finite group (2) Finitely generated periodic group (2) Finite group (2) |

Permutable implies subnormal in finite | | Finite group (?) Permutable subgroup (?) Subnormal subgroup (?) Permutable subgroup of finite group (2) Subnormal subgroup of finite group (3) |

Pyber's theorem on logarithmic quotient of number of nilpotent groups to number of groups approaching unity | | Finite group (?) Order of a group (?) Finite nilpotent group (?) |

Schur index of irreducible character in characteristic zero divides exponent | | Schur index of irreducible character (1) Irreducible linear representation (1) Exponent of a group (1) Finite group (1) Irreducible linear representation (?) Schur index of irreducible character (?) Exponent of a group (?) |

Square-free implies solvability-forcing | | Finite group (?) Solvable group (?) Finite solvable group (?) |

Subgroup of index equal to least prime divisor of group order is normal | | Subgroup of prime index (?) Index of a subgroup (?) Normal subgroup (?) Finite group (?) Subgroup of index equal to least prime divisor of group order (?) Subgroup of index equal to least prime divisor of group order of finite group (2) Normal subgroup of finite group (3) |

Sylow implies intermediately isomorph-conjugate | | Finite group (?) Sylow subgroup (?) Intermediately isomorph-conjugate subgroup (?) Sylow subgroup of finite group (2) Intermediately isomorph-conjugate subgroup of finite group (3) |

Sylow implies order-dominating | | Finite group (?) Sylow subgroup (?) Order-dominating subgroup (?) Sylow subgroup of finite group (2) Order-dominating subgroup of finite group (3) |

Sylow implies pronormal | | Finite group (?) Sylow subgroup (?) Pronormal subgroup (?) Sylow subgroup of finite group (2) Pronormal subgroup of finite group (3) |

Sylow normalizer implies abnormal | | Finite group (?) Sylow normalizer (?) Abnormal subgroup (?) Sylow normalizer of finite group (2) Abnormal subgroup of finite group (3) Sylow subgroup (?) Subgroup with abnormal normalizer (?) Sylow subgroup of finite group (2) Subgroup with abnormal normalizer of finite group (3) |

Sylow normalizer implies weakly abnormal | | Finite group (?) Sylow normalizer (?) Weakly abnormal subgroup (?) Sylow normalizer of finite group (2) Upward-closed self-normalizing subgroup of finite group (3) Self-normalizing subgroup (?) |

Sylow of normal implies pronormal | | Finite group (?) Sylow subgroup of normal subgroup (?) Pronormal subgroup (?) Sylow subgroup of normal subgroup of finite group (2) Pronormal subgroup of finite group (3) Composition operator (?) Sylow subgroup (?) Normal subgroup (?) |

Sylow subgroups exist | 3 | Finite group (3) Sylow subgroup (1) |

Sylow-permutable implies subnormal in finite | | Finite group (?) Sylow-permutable subgroup (?) Subnormal subgroup (?) Sylow-permutable subgroup of finite group (2) Subnormal subgroup of finite group (3) |

Upward-closed characteristic not implies cyclic-quotient in finite | | Finite group (2) Upward-closed characteristic subgroup (2) Cyclic-quotient subgroup (2) Characteristic subgroup (?) |