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 $G$ 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.
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 $\to$ 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.

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 $H$.

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

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

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

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