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]
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
Definition
A subgroup of a group is termed divisibilityclosed or divisibilityinvariant if it satisfies the following equivalent conditions:
 For every prime such that is divisible (i.e., for every , there exists such that ), is also divisible.
 For every natural number such that is divisible (i.e., for every , there exists such that ), is also divisible.
Note that we do not require that all the roots in of an element of must be in .
Metaproperties
Relation with other properties
Stronger properties
Property 
Meaning 
Proof of implication 
Proof of strictness (reverse implication failure) 
Intermediate notions

direct factor 
normal subgroup with normal complement 
(via retract) 


retract 
has a normal complement 


Endomorphism image, Local divisibilityclosed subgroup, Verbally closed subgroupFULL LIST, MORE INFO

subgroup of finite group 



Completely divisibilityclosed subgroupFULL LIST, MORE INFO

local divisibilityclosed subgroup 
if a particular element of the subgroup has a root in the whole group, the element has a root in the subgroup. 


FULL LIST, MORE INFO

completely divisibilityclosed subgroup 
divisibilityclosed, and all relevant roots lie inside the subgroup. 


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kernel of a bihomomorphism 
kernel of a bihomomorphism 
(via completely divisibilityclosed) 

Completely divisibilityclosed subgroupFULL LIST, MORE INFO

kernel of a multihomomorphism 

(via completely divisibilityclosed) 

Completely divisibilityclosed subgroupFULL LIST, MORE INFO

subgroup of finite index 

(via completely divisibilityclosed) 

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pure subgroup 
local divisibilityclosed subgroup of abelian group 
(via local divisibilityclosed) 
any example in a nonabelian group 
FULL LIST, MORE INFO

verbal subgroup of abelian group 

verbal subgroup of abelian group implies divisibilityclosed 

Divisibilityclosed subgroup of abelian groupFULL LIST, MORE INFO

Weaker properties
Satisfaction by subgroupdefining functions
Subgroupdefining function 
Always divisibilityclosed? 
Proof 
Stronger/weaker properties satisfied 
If the group property is restricted

center 
No 
center not is divisibilityclosed 
weaker: poweringinvariant subgroup (see center is poweringinvariant and center is local poweringinvariant) 
in nilpotent group, center is a completely divisibilityclosed subgroup (follows from upper central series members are completely divisibilityclosed in nilpotent group)

members of the upper central series 
No 
based on failure for center 
weaker: poweringinvariant subgroup (see upper central series members are poweringinvariant, upper central series members are quotientpoweringinvariant) 
in nilpotent group, they are divisibilityclosed subgroups, and in fact are completely divisibilityclosed subgroups (follows from upper central series members are completely divisibilityclosed in nilpotent group)

derived subgroup 
No 
derived subgroup not is divisibilityclosed 

in nilpotent group, yes (see derived subgroup is divisibilityclosed in nilpotent group, follows from equivalence of definitions of nilpotent group that is divisible for a set of primes)

members of the lower central series 
No 
based on failure for derived subgroup 

in nilpotent group, they are divisibilityclosed (see lower central series members are divisibilityclosed in nilpotent group, follows from equivalence of definitions of nilpotent group that is divisible for a set of primes)

members of the derived series 
No 
based on failure for derived subgroup 

in nilpotent group, they are divisibilityclosed subgroups. This follows from the result for the derived subgroup. See derived series members are divisibilityclosed in nilpotent group.

socle 
No 


Not true even in abelian groups, see monolith not is divisibilityclosed in abelian group. However, the weaker property of being a poweringinvariant subgroup is true in solvable groups: socle is poweringinvariant in solvable group
