This article describes a property that arises as the conjunction of a subgroup property: verbal 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
Suppose is an abelian group and is a subgroup of . We say that is a verbal subgroup of abelian group if it satisfies the following equivalent conditions:
 is a verbal subgroup of .
 There exists an integer such that is precisely the set of powers in .
Equivalence of definitions
Further information: verbal subgroup equals power subgroup in abelian group
Relation with other properties
Weaker properties
Property 
Meaning 
Proof of implication 
Proof of strictness (reverse implication failure) 
Intermediate notions

fully invariant subgroup of abelian group 


see fully invariant not implies verbal in finite abelian group 
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characteristic subgroup of abelian group 

(via fully invariant) 
(via fully invariant) 
Fully invariant subgroup of abelian groupFULL LIST, MORE INFO

verbal subgroup 



Abelian verbal subgroup, Verbal subgroup of nilpotent groupFULL LIST, MORE INFO

subgroup of abelian group 



Abelianpotentially verbal subgroup, Characteristic subgroup of abelian group, Fully invariant subgroup of abelian groupFULL LIST, MORE INFO

divisibilityclosed subgroup 

verbal subgroup of abelian group implies divisibilityclosed 

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poweringinvariant subgroup of abelian group 



Characteristic subgroup of abelian group, Divisibilityclosed subgroup of abelian groupFULL LIST, MORE INFO
