This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]
Definition
A subgroup of a group is termed an intermediately poweringinvariant subgroup if, for any subgroup of containing (i.e., ), is a poweringinvariant subgroup of .
Relation with other properties
Stronger properties
Property 
Meaning 
Proof of implication 
Proof of strictness (reverse implication failure) 
Intermediate notions

finite subgroup 

follows from finite implies poweringinvariant 

Intermediately local poweringinvariant subgroupFULL LIST, MORE INFO

periodic subgroup 



Intermediately local poweringinvariant subgroupFULL LIST, MORE INFO

subgroup of finite index 

follows from finite index implies poweringinvariant 

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normal subgroup of finite index 

(via subgroup of finite index) 
(via subgroup of finite index) 
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characteristic subgroup of abelian group 

characteristic subgroup of abelian group implies intermediately poweringinvariant 

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local divisibilityclosed subgroup 



Divisibilityclosed subgroup, Intermediately local poweringinvariant subgroup, Intermediately poweringinvariant subgroup, Local poweringinvariant subgroupFULL LIST, MORE INFO

retract 
has a normal complement 
(via local divisibilityclosed) 
(via local divisibilityclosed) 
Divisibilityclosed subgroup, Endomorphism image, Intermediately local poweringinvariant subgroup, Intermediately poweringinvariant subgroup, Local divisibilityclosed subgroup, Local poweringinvariant subgroup, Verbally closed subgroupFULL LIST, MORE INFO

complemented normal subgroup 
normal subgroup with a permutable complement, i.e., part of an internal semidirect product 
(via endomorphism kernel) 
(via endomorphism kernel) 
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direct factor 
factor in an internal direct product 
(via complemented normal, also via retract) 
(via complemented normal, also via retract) 
Intermediately local poweringinvariant subgroupFULL LIST, MORE INFO

intermediately local poweringinvariant subgroup 
local poweringinvariant subgroup in every intermediate subgroup 


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Weaker properties
Facts
Formalisms
In terms of the intermediately operator
This property is obtained by applying the intermediately operator to the property: poweringinvariant subgroup
View other properties obtained by applying the intermediately operator