Cofactorial automorphism-invariant subgroup
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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 a cofactorial automorphism-invariant subgroup if it is invariant under all the cofactorial automorphisms of the whole group.
For a finite group
Suppose is a finite group. A subgroup
of
is termed a cofactorial automorphism-invariant subgroup if
is invariant under every automorphism
of
for which all prime factors of the order of
are prime factors of the order of
.
For a periodic group
Suppose is a periodic group: every element of
has finite order. A subgroup
of
is termed cofactorial automorphism-invariant if
is invariant under every automorphism
of
such that
has finite order, and every prime divisor of the order of
occurs as the prime divisor of the order of some element of
.
For a general group
If the group has any element of infinite order, we define cofactorial automorphism-invariant to be the same as characteristic subgroup.
For a p-group
If is a p-group for some prime
(in the finite case, this means
is a group of prime power order, in the infinite case it simply means that every element has order a power of
), a cofactorial automorphism-invariant subgroup is the same as a p-automorphism-invariant subgroup.
Formalisms
Function restriction expression
This subgroup property is a function restriction-expressible subgroup property: it can be expressed by means of the function restriction formalism, viz there is a function restriction expression for it.
Find other function restriction-expressible subgroup properties | View the function restriction formalism chart for a graphic placement of this property
- The property of being a cofactorial automorphism-invariant subgroup can be expressed as an invariance property:
Cofactorial automorphism Function
In other words, is cofactorial automorphism-invariant in
if every cofactorial automorphism of
restricts to a function from
to itself.
- This can be written as an endo-invariance property:
Cofactorial automorphism Endomorphism
In other words, is cofactorial automorphism-invariant in
if every cofactorial automorphism of
restricts to an endomorphism of
.
- Since the inverse of an automorphism has the same order as that automorphism, and restriction of automorphism to subgroup invariant under it and its inverse is automorphism this can be further strengthened to an auto-invariance property:
Cofactorial automorphism Automorphism
In other words, is cofactorial automorphism-invariant in
if every cofactorial automorphism of
restricts to an automorphism of
.
Examples
VIEW: subgroups of groups satisfying this property | subgroups of groups dissatisfying this property
VIEW: Related subgroup property satisfactions | Related subgroup property dissatisfactions
Relation with other properties
Stronger properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
characteristic subgroup | invariant under all automorphisms | (direct) | follows from p-automorphism-invariant not implies characteristic | |FULL LIST, MORE INFO |
p-automorphism-invariant subgroup in a p-group | (direct) | (any group other than a p-group) |
Weaker properties
Related properties
Metaproperties
BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)
Here is a summary:
Metaproperty name | Satisfied? | Proof | Difficulty level (0-5) | Statement with symbols |
---|---|---|---|---|
transitive subgroup property | No | cofactorial automorphism-invariance is not transitive | It is possible to have groups ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
trim subgroup property | Yes | For any group ![]() ![]() ![]() | ||
intermediate subgroup condition | No | Can use examples for characteristicity does not satisfy intermediate subgroup condition | It is possible to have groups ![]() ![]() ![]() ![]() | |
strongly intersection-closed subgroup property | Yes | follows from invariance implies strongly intersection-closed | If ![]() ![]() ![]() | |
strongly join-closed subgroup property | Yes | follows from endo-invariance implies strongly join-closed | If ![]() ![]() ![]() |
Effect of property operators
The left transiter
Applying the left transiter to this property gives: characteristic subgroup