Groupprops, The Group Properties Wiki (pre-alpha)

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Contents 1 Definition 1.1 For a finite group 1.2 For a periodic group 1.3 For a general group 1.4 For a p-group 2 Formalisms 2.1 Function restriction expression 3 Examples 4 Relation with other properties 4.1 Stronger properties 4.2 Weaker properties 4.3 Related properties 5 Metaproperties 5.1 Transitivity 5.2 Intersection-closedness 5.3 Join-closedness 6 Effect of property operators 6.1 The left transiter

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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.
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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 G is a finite group. A subgroup H of G is termed a cofactorial automorphism-invariant subgroup if H is invariant under every automorphism σ of G for which all prime factors of the order of σ are prime factors of the order of G.

For a periodic group

Suppose G is a periodic group: every element of G has finite order. A subgroup H of G is termed cofactorial automorphism-invariant if H is invariant under every automorphism σ of G such that σ has finite order, and every prime divisor of the order of σ occurs as the prime divisor of the order of some element of G.

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 G is a p-group for some prime p (in the finite case, this means G is a group of prime power order, in the infinite case it simply means that every element has order a power of p), 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, H is cofactorial automorphism-invariant in G if every cofactorial automorphism of G restricts to a function from H to itself.

• This can be written as an endo-invariance property:

Cofactorial automorphism Endomorphism

In other words, H is cofactorial automorphism-invariant in G if every cofactorial automorphism of G restricts to an endomorphism of H.

• 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, H is cofactorial automorphism-invariant in G if every cofactorial automorphism of G restricts to an automorphism of H.

Examples

VIEW: | subgroups of groups dissatisfying this property
VIEW: |

Relation with other properties

Stronger properties

• Characteristic subgroup
• Subnormal stability automorphism-invariant subgroup
• p-automorphism-invariant subgroup in a p-group

Weaker properties

• Sub-cofactorial automorphism-invariant subgroup
• Subgroup-cofactorial automorphism-invariant subgroup
• Left-transitively 2-subnormal subgroup: For full proof, refer: Cofactorial automorphism-invariant implies left-transitively 2-subnormal
• Normal subgroup: For full proof, refer: Cofactorial automorphism-invariant implies normal. Also related:
  • Normal subgroup of characteristic subgroup
  • 2-subnormal subgroup

Related properties

• Coprime automorphism-invariant subgroup
• Coprime automorphism-invariant normal subgroup

Metaproperties

Transitivity

NO: This subgroup property is not transitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole group
ABOUT THIS PROPERTY: |
ABOUT TRANSITIVITY: View a complete list of subgroup properties that are not transitive| View facts related to transitivity of subgroup properties | View a survey article on disproving transitivity

A cofactorial automorphism-invariant subgroup of a cofactorial automorphism-invariant subgroup need not be cofactorial automorphism-invariant. This is because the prime factors of the order of the subgroup may be considerably fewer than those of the order of the whole group. For full proof, refer: Cofactorial automorphism-invariance is not transitive

Intersection-closedness

YES: This subgroup property is intersection-closed: an arbitrary (nonempty) intersection of subgroups with this property, also has this property.
ABOUT THIS PROPERTY: |
ABOUT INTERSECTION-CLOSEDNESS: View all intersection-closed subgroup properties (or, strongly intersection-closed properties) | View all subgroup properties that are not intersection-closed | Read a survey article on proving intersection-closedness | Read a survey article on disproving intersection-closedness

An arbitrary intersection of cofactorial automorphism-invariant subgroups is again cofactorial automorphism-invariant. For full proof, refer: Cofactorial automorphism-invariance is strongly intersection-closed, Invariance implies strongly intersection-closed

Join-closedness

YES: This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property.
ABOUT THIS PROPERTY: |
ABOUT JOIN-CLOSEDNESS: View all join-closed subgroup properties (or, strongly join-closed properties) | View all subgroup properties that are not join-closed | Read a survey article on proving join-closedness | Read a survey article on disproving join-closedness

An arbitrary join of cofactorial automorphism-invariant subgroups is again cofactorial automorphism-invariant. This follows from the fact that cofactorial automorphism-invariance is an endo-invariance property. For full proof, refer: Cofactorial automorphism-invariance is strongly join-closed, Endo-invariance implies strongly join-closed

Effect of property operators

The left transiter

Applying the left transiter to this property gives: characteristic subgroup