# Semantic search

Abelian Frattini subgroup implies centralizer is critical, Abelian and abelian automorphism group not implies locally cyclic, Abelian and ambivalent iff elementary abelian 2-group, Abelian automorphism group not implies abelian, Abelian implies every element is automorphic to its inverse, Abelian implies every subgroup is potentially characteristic, Abelian implies nilpotent, Abelian implies self-centralizing in holomorph, Abelian normal is not join-closed, Abelianness is 2-local, Abelianness is directed union-closed, Abelianness is quotient-closed, Abelianness is subgroup-closed, CA not implies nilpotent, Characteristic subgroup of abelian group not implies divisibility-closed, Characteristic subgroup of abelian group not implies local powering-invariant, Characteristic subgroup of uniquely p-divisible abelian group is uniquely p-divisible, Commuting fraction more than five-eighths implies abelian, Cube map is endomorphism iff abelian (if order is not a multiple of 3), Cyclic automorphism group implies abelian, Cyclic implies abelian, Dedekind not implies abelian, Exponent two implies abelian, Finite abelian and abelian automorphism group implies cyclic, Finite abelian implies same orbit sizes of conjugacy classes and irreducible representations under automorphism group, Full invariance is not direct power-closed, Fully invariant not implies abelian-potentially verbal in abelian group, Fully invariant subgroup of abelian group not implies divisibility-closed, Intermediately characteristic not implies isomorph-containing in abelian group, Nilpotent not implies abelian, Nonempty characteristic subsemigroup of abelian group implies subgroup, Quotients of elementarily equivalent abelian groups by multiples of n are elementarily equivalent, Rational and abelian implies elementary abelian 2-group, There exist abelian groups whose isomorphism classes of direct powers have any given period, Torsion subgroups of elementary equivalent abelian groups are elementarily equivalent, Verbal subgroup of abelian group not implies local powering-invariant