This article describes a property that arises as the conjunction of a subgroup property: characteristic subgroup with a group property imposed on the ambient group: abelian group
View a complete list of such conjunctions  View a complete list of conjunctions where the group property is imposed on the subgroup
Definition
A subgroup of a group is termed a characteristic subgroup of abelian group if is a characteristic subgroup of and is an abelian group.
Relation with other properties
Stronger properties
Weaker properties
Property 
Meaning 
Proof of implication 
Proof of strictness (reverse implication failure) 
Intermediate notions

abelian characteristic subgroup 
the subgroup is characteristic and abelian as a group in its own right; the whole group need not be abelian. 
Follows from abelianness is subgroupclosed 
the trivial subgroup in any nontrivial group. 
Characteristic central subgroup, Characteristic subgroup of centerFULL LIST, MORE INFO

characteristic central subgroup 
characteristic subgroup that is contained in the center 
obvious 
trivial subgroup in nonabelian group 
Characteristic subgroup of centerFULL LIST, MORE INFO

subgroup of abelian group 
the whole group is abelian; no restriction on the subgroup. 
obvious 
any noncharacteristic subgroup of an abelian group, e.g., a cyclic subgroup in an elementary abelian group on two or more generators. 
Abelianextensible automorphisminvariant subgroup, Abelianpotentially characteristic subgroup, Abelianquotientpullbackable automorphisminvariant subgroupFULL LIST, MORE INFO

characteristic subgroup of center 
the subgroup is contained in the center and is characteristic as a subgroup of the center 

trivial subgroup in nonabelian group 
FULL LIST, MORE INFO

poweringinvariant subgroup and also poweringinvariant subgroup of abelian group 
if the group 
characteristic subgroup of abelian group implies poweringinvariant 
(any noncharacteristic subgroup of a finite abelian group) 
Characteristic subgroup of center, Intermediately poweringinvariant subgroup, Poweringinvariant characteristic subgroup, Poweringinvariant normal subgroup, Poweringinvariant subgroup of abelian group, Quotientpoweringinvariant characteristic subgroup, Quotientpoweringinvariant subgroupFULL LIST, MORE INFO

characteristic subgroup of nilpotent group 
the subgroup is characteristic and the whole group is a nilpotent group 
follows from abelian implies nilpotent 
follows from nilpotent not implies abelian (we can use the whole group as the subgroup of itself) 
FULL LIST, MORE INFO
