# Cofactorial automorphism-invariant subgroup

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BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
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