Cofactorial automorphism-invariant subgroup

Definition
A subgroup of a group is termed a cofactorial automorphism-invariant subgroup if it is invariant under all the defining ingredient::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 $$\sigma$$ of $$G$$ for which all prime factors of the order of $$\sigma$$ 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 $$\sigma$$ of $$G$$ such that $$\sigma$$ has finite order, and every prime divisor of the order of $$\sigma$$ 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 defining ingredient::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

 * The property of being a cofactorial automorphism-invariant subgroup can be expressed as an :

Cofactorial automorphism $$\to$$ 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 $$\to$$ 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 $$\to$$ Automorphism

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

Stronger properties

 * Weaker than::Characteristic subgroup
 * Weaker than::Subnormal stability automorphism-invariant subgroup
 * Weaker than::p-automorphism-invariant subgroup in a p-group

Weaker properties

 * Stronger than::Sub-cofactorial automorphism-invariant subgroup
 * Stronger than::Subgroup-cofactorial automorphism-invariant subgroup
 * Stronger than::Left-transitively 2-subnormal subgroup:
 * Stronger than::Normal subgroup: . Also related:
 * Stronger than::Normal subgroup of characteristic subgroup
 * Stronger than::2-subnormal subgroup

Related properties

 * Coprime automorphism-invariant subgroup
 * Coprime automorphism-invariant normal subgroup

Metaproperties
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.

An arbitrary intersection of cofactorial automorphism-invariant subgroups is again cofactorial automorphism-invariant.

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.