Local powering-invariant subgroup
From Groupprops
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 a local powering-invariant subgroup if the following hold:
- Whenever
and
are such that there is a unique
such that
, we must have
.
- Whenever
and
is a prime number such that there is a unique
such that
, we must have
.
Equivalence of definitions
Further information: equivalence of definitions of local powering-invariant subgroup
Metaproperties
Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|
transitive subgroup property | Yes | local powering-invariance is transitive | Suppose ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
strongly intersection-closed subgroup property | Yes | local powering-invariance is strongly intersection-closed | Suppose ![]() ![]() ![]() ![]() |
intermediate subgroup condition | No | local powering-invariance does not satisfy intermediate subgroup condition | It is possible to have groups ![]() ![]() ![]() ![]() |
finite-join-closed subgroup property | No | local powering-invariance is not finite-join-closed | It is possible to have a group ![]() ![]() ![]() ![]() ![]() |
centralizer-closed subgroup property | Yes | follows from c-closed implies local powering-invariant | Suppose ![]() ![]() ![]() ![]() ![]() ![]() |
commutator-closed subgroup property | No | local powering-invariance is not commutator-closed | It is possible to have a group ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
image condition | No | local powering-invariance does not satisfy image condition | It is possible to have a group ![]() ![]() ![]() ![]() ![]() ![]() |
Relation with other properties
Stronger properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
finite subgroup | finite implies local powering-invariant | |||
periodic subgroup | ||||
fixed-point subgroup of a subgroup of the automorphism group | fixed-point subgroup of a subgroup of the automorphism group implies local powering-invariant | |||
c-closed subgroup | (via fixed-point subgroup of a subgroup of the automorphism group) | |||
local divisibility-closed subgroup | |FULL LIST, MORE INFO | |||
retract | has a normal complement | (via local divisibility-closed) | (via local divisibility-closed) | Intermediately local powering-invariant subgroup, Local divisibility-closed subgroup, Verbally closed subgroup|FULL LIST, MORE INFO |
direct factor | normal subgroup with normal complement | (via retract) | (via retract) | Intermediately local powering-invariant subgroup, Local divisibility-closed subgroup, Retract, Verbally closed subgroup|FULL LIST, MORE INFO |
Weaker properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
powering-invariant subgroup |
Incomparable properties
Property | Meaning | Proof that it does not imply being local powering-invariant | Proof that being local powering-invariant does not imply it | Properties stronger than both | Properties weaker than both |
---|---|---|---|---|---|
subgroup of finite index | has finite index in the whole group | finite index not implies local powering-invariant | any direct factor where the other direct factor is infinite. | |FULL LIST, MORE INFO | Powering-invariant subgroup|FULL LIST, MORE INFO |
complemented normal subgroup | normal subgroup with a permutable complement | complemented normal not implies local powering-invariant | any non-normal subgroup of a finite group will do. | Direct factor|FULL LIST, MORE INFO | Powering-invariant subgroup|FULL LIST, MORE INFO |