Abelian Frattini subgroup implies centralizer is critical | Frattini subgroup is characteristic Characteristicity is centralizer-closed | Frattini subgroup (?) Abelian group (?) Critical subgroup (?) C-closed critical subgroup (?) |

Abelian and abelian automorphism group not implies locally cyclic | | Abelian group (3) Group whose automorphism group is abelian (2) Locally cyclic group (2) |

Abelian and ambivalent iff elementary abelian 2-group | | Abelian group (3) Ambivalent group (2) Elementary abelian 2-group (3) |

Abelian automorphism group not implies abelian | | Group whose automorphism group is abelian (2) Abelian group (2) |

Abelian implies every element is automorphic to its inverse | Inverse map is automorphism iff abelian | Abelian group (2) Group in which every element is automorphic to its inverse (2) |

Abelian implies every subgroup is potentially characteristic | Central implies potentially characteristic Abelian implies every subgroup is central | Abelian group (?) Subgroup (?) Potentially characteristic subgroup (?) Subgroup of abelian group (2) Potentially characteristic subgroup of abelian group (3) |

Abelian implies nilpotent | | Abelian group (2) Nilpotent group (2) |

Abelian implies self-centralizing in holomorph | | Abelian group (2) NSCFN-realizable group (2) Holomorph of a group (?) Self-centralizing subgroup (?) NSCFN-subgroup (?) NSCFN-realizable group (?) |

Abelian normal is not join-closed | | Abelian normal subgroup (1) Join-closed subgroup property (2) Abelian group (1) Normal join-closed group property (2) |

Abelianness is 2-local | | Abelian group (1) 2-local group property (2) |

Abelianness is directed union-closed | Abelianness is 2-local Local implies directed union-closed | Abelian group (1) Directed union-closed group property (2) |

Abelianness is quotient-closed | | Abelian group (1) Quotient-closed group property (2) |

Abelianness is subgroup-closed | | Abelian group (1) Subgroup-closed group property (2) |

CA not implies nilpotent | | CA-group (2) Nilpotent group (2) CN-group (2) Abelian group (3) |

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 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 (?) |

Commuting fraction more than five-eighths implies abelian | Cyclic over central implies abelian Lagrange's theorem Subgroup of size more than half is whole group | Commuting fraction (2) Finite abelian group (?) Number of conjugacy classes (2) FZ-group (2) Abelian group (3) |

Cube map is endomorphism iff abelian (if order is not a multiple of 3) | Abelian implies universal power map is endomorphism Cube map is surjective endomorphism implies abelian Kth power map is bijective iff k is relatively prime to the order | Cube map (?) Abelian group (?) |

Cyclic automorphism group implies abelian | Group acts as automorphisms by conjugation Cyclicity is subgroup-closed Cyclic over central implies abelian | Group whose automorphism group is cyclic (?) Automorphism group of a group (?) Cyclic group (?) Abelian group (?) |

Cyclic implies abelian | | Cyclic group (2) Abelian group (2) |

Dedekind not implies abelian | | Dedekind group (2) Abelian group (2) |

Exponent two implies abelian | Square map is endomorphism iff abelian | Exponent of a group (2) Abelian group (3) Elementary abelian group (2) |

Finite abelian and abelian automorphism group implies cyclic | | Finite abelian group (3) Group whose automorphism group is abelian (2) Abelian group (3) Cyclic group (3) Finite cyclic groupl3 (?) |

Finite abelian implies same orbit sizes of conjugacy classes and irreducible representations under automorphism group | | Finite abelian group (2) Finite group having the same orbit sizes of conjugacy classes and irreducible representations under automorphism group (2) Finite abelian group (?) Abelian group (?) |

Full invariance is not direct power-closed | | Fully invariant subgroup (1) Direct power-closed subgroup property (2) Abelian group (?) |

Fully invariant not implies abelian-potentially verbal in abelian group | Divisible abelian subgroup of abelian group contains no proper nontrivial verbal subgroup | Abelian group (2) Fully invariant subgroup (2) Abelian-potentially verbal subgroup (2) |

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

Intermediately characteristic not implies isomorph-containing in abelian group | | Abelian group (2) Intermediately characteristic subgroup (2) Isomorph-containing subgroup (2) |

Nilpotent not implies abelian | | Nilpotent group (2) Abelian group (2) |

Nonempty characteristic subsemigroup of abelian group implies subgroup | Inverse map is automorphism iff abelian | Abelian group (3) Characteristic subgroup (3) |

Quotients of elementarily equivalent abelian groups by multiples of n are elementarily equivalent | | Abelian group (?) Elementarily equivalent groups (?) |

Rational and abelian implies elementary abelian 2-group | | Rational group (2) Abelian group (3) |

There exist abelian groups whose isomorphism classes of direct powers have any given period | | Group isomorphic to its cube (2) Group isomorphic to its square (2) Torsion-free abelian group (?) Abelian group (?) Torsion-free group (?) Direct power of a group (?) Isomorphic groups (?) Group isomorphic to its cube (?) Group isomorphic to its square (?) |

Torsion subgroups of elementary equivalent abelian groups are elementarily equivalent | | Abelian group (?) Elementarily equivalent groups (?) Torsion subgroup (?) |

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