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]

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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 $\sigma$ 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 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.
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• The property of being a cofactorial automorphism-invariant subgroup can be expressed as an invariance property:

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 $H$.

Examples

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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

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
subnormal stability automorphism-invariant subgroup	invariant under any stability automorphism of any subnormal series			|FULL LIST, MORE INFO
sub-cofactorial automorphism-invariant subgroup	there is a series from the subgroup to the whole group where each is a cofactorial automorphism-invariant subgroup in its successor.	(direct)	(involves cases where the set of prime divisors shrinks as we go from the group down to the subgroup)	|FULL LIST, MORE INFO
subgroup-cofactorial automorphism-invariant subgroup	invariant under any automorphism whose order has no prime factors other than those of the order of the subgroup.	(direct)		Sub-cofactorial automorphism-invariant subgroup|FULL LIST, MORE INFO
left-transitively 2-subnormal subgroup	whenever the whole group is 2-subnormal in a bigger group, so is the subgroup.	Cofactorial automorphism-invariant implies left-transitively 2-subnormal		Sub-cofactorial automorphism-invariant subgroup, Subgroup-cofactorial automorphism-invariant subgroup|FULL LIST, MORE INFO
normal subgroup	invariant under inner automorphisms	cofactorial automorphism-invariant implies normal		|FULL LIST, MORE INFO

Related properties

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

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 $H \le K \le G$ such that $H$ is cofactorial automorphism-invariant in $K$ and $K$ is cofactorial automorphism-invariant in $G$ but $H$ is not cofactorial automorphism-invariant in $G$.
trim subgroup property	Yes			For any group $G$, the subgroups $\{ e \}$ and $G$ 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 $H \le K \le G$ such that $H$ is cofactorial automorphism-invariant in $G$ but not in $K$.
strongly intersection-closed subgroup property	Yes	follows from invariance implies strongly intersection-closed		If $H_i, i \in I$ are all cofactorial automorphism-invariant subgroups of a group $G$, so is the intersection of subgroups $\bigcap_{i \in I} H_i$.
strongly join-closed subgroup property	Yes	follows from endo-invariance implies strongly join-closed		If $H_i, i \in I$ are all cofactorial automorphism-invariant subgroups of a group $G$, so is the join of subgroups $\left \langle H_i \right \rangle_{i \in I}$.

Effect of property operators

The left transiter

Applying the left transiter to this property gives: characteristic subgroup