AEP upper-hook characteristic implies AEP | | AEP-subgroup (?) Characteristic subgroup (?) |

Additive group of a field implies characteristic in holomorph | Additive group of a field implies monolith in holomorph Monolith is characteristic | Additive group of a field (2) Holomorph-characteristic group (2) Additive group of a field (?) Characteristically simple group (?) Elementary abelian group (?) Characteristic subgroup (?) Holomorph of a group (?) |

Analogue of critical subgroup theorem for finite solvable groups | | Finite solvable group (?) Characteristic subgroup (?) Frattini-in-center group (?) Self-centralizing subgroup (?) Coprime automorphism-faithful subgroup (?) |

Center is characteristic | | Center (1) Characteristic subgroup (2) |

Centerless and characteristic in automorphism group implies automorphism group is complete | Centerless implies inner automorphism group is centralizer-free in automorphism group Inner automorphism group is normal in automorphism group Normal and centralizer-free implies automorphism-faithful Automorphism group action lemma | Centerless group (?) Automorphism group of a group (?) Characteristic subgroup (?) Complete group (?) |

Characteristic and self-centralizing implies coprime automorphism-faithful | | Self-centralizing characteristic subgroup (2) Coprime automorphism-faithful subgroup (2) Characteristic subgroup (?) Self-centralizing subgroup (?) Coprime automorphism-faithful characteristic subgroup (?) |

Characteristic central factor of WNSCDIN implies WNSCDIN | Characteristic of normal implies normal | Composition operator (?) Characteristic central factor (?) WNSCDIN-subgroup (?) Characteristic central factor (2) Left-transitively WNSCDIN-subgroup (2) Characteristic subgroup (?) Central factor (?) |

Characteristic equals fully invariant in odd-order abelian group | | Odd-order abelian group (?) Characteristic subgroup (?) Fully invariant subgroup (?) Characteristic subgroup of odd-order abelian group (2) Fully invariant subgroup of odd-order abelian group (3) |

Characteristic equals strictly characteristic in Hopfian | | Hopfian group (?) Characteristic subgroup (?) Strictly characteristic subgroup (?) Characteristic subgroup of Hopfian group (2) Strictly characteristic subgroup of Hopfian group (3) |

Characteristic equals verbal in free abelian group | | Free abelian group (?) Characteristic subgroup (?) Verbal subgroup (?) Characteristic subgroup of free abelian group (2) Verbal subgroup of free abelian group (3) |

Characteristic implies automorph-conjugate | | Characteristic subgroup (2) Automorph-conjugate subgroup (2) |

Characteristic implies normal | Group acts as automorphisms by conjugation | Characteristic subgroup (1) Normal subgroup (1) |

Characteristic maximal subgroups may be isomorphic and distinct in group of prime power order | | Characteristic subgroup (?) Maximal subgroup of group of prime power order (?) Series-equivalent subgroups (?) |

Characteristic not implies amalgam-characteristic | | Characteristic subgroup (2) Amalgam-characteristic subgroup (1) |

Characteristic not implies characteristic-isomorph-free in finite | | Characteristic subgroup (2) Characteristic-isomorph-free subgroup (2) |

Characteristic not implies direct factor | | Characteristic subgroup (2) Direct factor (2) |

Characteristic not implies elementarily characteristic | | Characteristic subgroup (2) Elementarily characteristic subgroup (2) |

Characteristic not implies fully invariant | | Characteristic subgroup (1) Fully invariant subgroup (1) |

Characteristic not implies fully invariant in class three maximal class p-group | | Maximal class group (?) Characteristic subgroup (?) Fully invariant subgroup (?) |

Characteristic not implies fully invariant in finite abelian group | | Finite abelian group (2) Characteristic subgroup (2) Fully invariant subgroup (2) |

Characteristic not implies fully invariant in finitely generated abelian group | | Finitely generated abelian group (2) Characteristic subgroup (2) Fully invariant subgroup (2) |

Characteristic not implies fully invariant in odd-order class two p-group | | Odd-order class two p-group (2) Characteristic subgroup (2) Fully invariant subgroup (2) |

Characteristic not implies injective endomorphism-invariant | Finitary symmetric group is characteristic in symmetric group Finitary symmetric group is not injective endomorphism-invariant in symmetric group Center is characteristic Center not is injective endomorphism-invariant | Characteristic subgroup (2) Injective endomorphism-invariant subgroup (2) |

Characteristic not implies injective endomorphism-invariant in finitely generated abelian group | | Finitely generated abelian group (2) Characteristic subgroup (2) Injective endomorphism-invariant subgroup (2) |

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 | Characteristic not implies characteristic-isomorph-free in finite Normal-isomorph-free implies characteristic-isomorph-free Characteristic-isomorph-free not implies normal-isomorph-free in finite Characteristic-isomorph-free implies characteristic Series-equivalent characteristic subgroups may be distinct Normal-isomorph-free implies series-isomorph-free | Finite group (2) Characteristic subgroup (2) Normal-isomorph-free subgroup (2) |

Characteristic not implies potentially fully invariant | Normal not implies potentially fully invariant Normal equals potentially characteristic | Characteristic subgroup (2) Potentially fully invariant subgroup (2) Fully invariant subgroup (3) |

Characteristic not implies powering-invariant in solvable group | | Solvable group (2) Characteristic subgroup (2) Powering-invariant subgroup (2) |

Characteristic not implies quasiautomorphism-invariant | | Characteristic subgroup (2) Quasiautomorphism-invariant subgroup (2) |

Characteristic not implies strictly characteristic | | Characteristic subgroup (2) Strictly characteristic 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) |

Characteristic of CDIN implies CDIN | | Composition operator (?) Characteristic subgroup (?) CDIN-subgroup (?) Characteristic subgroup (2) Left-transitively CDIN-subgroup (2) |

Characteristic of normal implies normal | Restriction of automorphism to subgroup invariant under it and its inverse is automorphism Composition rule for function restriction | Composition operator (?) Characteristic subgroup (?) Normal subgroup (?) |

Characteristic of potentially characteristic implies potentially characteristic | | Composition operator (?) Characteristic subgroup (?) Potentially characteristic subgroup (?) |

Characteristic subgroup of Sylow subgroup is weakly closed iff it is normal in every Sylow subgroup containing it | Weakly closed implies conjugation-invariantly relatively normal in finite group Sylow implies order-conjugate Sylow implies WNSCDIN WNSCDIN implies every normalizer-relatively normal conjugation-invariantly relatively normal subgroup is weakly closed | Sylow subgroup (?) Characteristic subgroup (?) Weakly closed subgroup (?) |

Characteristic subgroup of abelian group not implies divisibility-closed | | Characteristic subgroup of abelian group (2) Divisibility-closed subgroup (2) Abelian group (2) Characteristic subgroup (2) |

Characteristic subgroup of abelian group not implies local powering-invariant | | Characteristic subgroup of abelian group (2) Local powering-invariant subgroup (2) Abelian group (2) Characteristic subgroup (2) |

Characteristic subgroup of additive group of odd-order Lie ring is derivation-invariant and fully invariant | Characteristic equals fully invariant in odd-order abelian group Fully invariant subgroup of additive group of Lie ring is derivation-invariant and fully invariant | Characteristic subgroup (?) Derivation-invariant Lie subring (?) Ideal of a Lie ring (?) Fully invariant Lie subring (?) |

Characteristic subgroup of uniquely p-divisible abelian group is uniquely p-divisible | Abelian implies uniquely p-divisible iff pth power map is automorphism | Abelian group (?) Characteristic subgroup (?) Powered group for a set of primes (?) |

Characteristic subset generates characteristic subgroup | | Characteristic subset of a group (1) Characteristic subgroup (3) |

Characteristic upper-hook AEP implies characteristic | | AEP-subgroup (2) Subgroup in which every subgroup characteristic in the whole group is characteristic (2) Characteristic subgroup (?) AEP-subgroup (?) |

Characteristicity does not satisfy image condition | | Characteristic subgroup (1) Image condition (2) |

Characteristicity does not satisfy intermediate subgroup condition | | Characteristic subgroup (1) Intermediate subgroup condition (2) |

Characteristicity does not satisfy lower central series condition | | Characteristic subgroup (1) Lower central series condition (2) |

Characteristicity is centralizer-closed | | Characteristic subgroup (1) Centralizer-closed subgroup property (2) |

Characteristicity is commutator-closed | | Characteristic subgroup (1) Commutator-closed subgroup property (2) |

Characteristicity is not finite direct power-closed | | Characteristic subgroup (1) Finite direct power-closed subgroup property (2) |

Characteristicity is not finite-relative-intersection-closed | | Characteristic subgroup (1) Finite-relative-intersection-closed subgroup property (2) |