| Abelian Frattini subgroup implies centralizer is critical | | 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 | | Abelian group (2)
Group in which every element is automorphic to its inverse (2) |
| Abelian implies every subgroup is potentially characteristic | | 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 | | 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 group (?)
Characteristic subgroup (?)
Powered group for a set of primes (?) |
| Commuting fraction more than five-eighths implies abelian | | 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) | 3 | Cube map (?)
Abelian group (?) |
| Cyclic automorphism group 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 | | 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 | | 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 | | 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) |
