# 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 propertyVIEW: 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 such that is cofactorial automorphism-invariant in and is cofactorial automorphism-invariant in but is not cofactorial automorphism-invariant in . | |

trim subgroup property | Yes | For any group , the subgroups and are cofactorial automorphism-invariant. | ||

intermediate subgroup condition | No | Can use examples for characteristicity does not satisfy intermediate subgroup condition | It is possible to have groups such that is cofactorial automorphism-invariant in but not in . | |

strongly intersection-closed subgroup property | Yes | follows from invariance implies strongly intersection-closed | If are all cofactorial automorphism-invariant subgroups of a group , so is the intersection of subgroups . | |

strongly join-closed subgroup property | Yes | follows from endo-invariance implies strongly join-closed | If are all cofactorial automorphism-invariant subgroups of a group , so is the join of subgroups . |

## Effect of property operators

### The left transiter

*Applying the left transiter to this property gives*: characteristic subgroup