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 divisibility-closed or divisibility-invariant 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 divisibility-closed subgroup, Verbally closed subgroup|FULL LIST, MORE INFO
|
subgroup of finite group |
|
|
|
Completely divisibility-closed subgroup|FULL LIST, MORE INFO
|
local divisibility-closed 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 divisibility-closed subgroup |
divisibility-closed, and all relevant roots lie inside the subgroup. |
|
|
|FULL LIST, MORE INFO
|
kernel of a bihomomorphism |
kernel of a bihomomorphism |
(via completely divisibility-closed) |
|
Completely divisibility-closed subgroup|FULL LIST, MORE INFO
|
kernel of a multihomomorphism |
|
(via completely divisibility-closed) |
|
Completely divisibility-closed subgroup|FULL LIST, MORE INFO
|
subgroup of finite index |
|
(via completely divisibility-closed) |
|
|FULL LIST, MORE INFO
|
pure subgroup |
local divisibility-closed subgroup of abelian group |
(via local divisibility-closed) |
any example in a non-abelian group |
|FULL LIST, MORE INFO
|
verbal subgroup of abelian group |
|
verbal subgroup of abelian group implies divisibility-closed |
|
Divisibility-closed subgroup of abelian group|FULL LIST, MORE INFO
|
Weaker properties
Satisfaction by subgroup-defining functions
Subgroup-defining function |
Always divisibility-closed? |
Proof |
Stronger/weaker properties satisfied |
If the group property is restricted
|
center |
No |
center not is divisibility-closed |
weaker: powering-invariant subgroup (see center is powering-invariant and center is local powering-invariant) |
in nilpotent group, center is a completely divisibility-closed subgroup (follows from upper central series members are completely divisibility-closed in nilpotent group)
|
members of the upper central series |
No |
based on failure for center |
weaker: powering-invariant subgroup (see upper central series members are powering-invariant, upper central series members are quotient-powering-invariant) |
in nilpotent group, they are divisibility-closed subgroups, and in fact are completely divisibility-closed subgroups (follows from upper central series members are completely divisibility-closed in nilpotent group)
|
derived subgroup |
No |
derived subgroup not is divisibility-closed |
|
in nilpotent group, yes (see derived subgroup is divisibility-closed 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 divisibility-closed (see lower central series members are divisibility-closed 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 divisibility-closed subgroups. This follows from the result for the derived subgroup. See derived series members are divisibility-closed in nilpotent group.
|
socle |
No |
|
|
Not true even in abelian groups, see monolith not is divisibility-closed in abelian group. However, the weaker property of being a powering-invariant subgroup is true in solvable groups: socle is powering-invariant in solvable group
|