# Sub-cofactorial automorphism-invariant subgroup

From Groupprops

## Contents

## Definition

A subgroup of a group is termed a **sub-cofactorial automorphism-invariant subgroup** if there exists an ascending chain of subgroups:

such that each is a cofactorial automorphism-invariant subgroup of .

## Formalisms

### In terms of the subordination operator

This property is obtained by applying the subordination operator to the property: cofactorial automorphism-invariant subgroup

View other properties obtained by applying the subordination operator

## 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 | Cofactorial automorphism-invariant subgroup|FULL LIST, MORE INFO | ||

cofactorial automorphism-invariant subgroup | invariant under all cofactorial automorphisms | cofactorial automorphism-invariance is not transitive | |FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Subgroup-cofactorial automorphism-invariant subgroup | invariant under all subgroup-cofactorial automorphisms | |FULL LIST, MORE INFO | ||

left-transitively 2-subnormal subgroup | whenever the group is 2-subnormal in a bigger group, so is the subgroup | Subgroup-cofactorial automorphism-invariant subgroup|FULL LIST, MORE INFO | ||

2-subnormal subgroup | normal subgroup of a normal subgroup | Left-transitively 2-subnormal subgroup, Subgroup-cofactorial automorphism-invariant subgroup|FULL LIST, MORE INFO | ||

subnormal subgroup | there is a subnormal series from it to the whole group | Subgroup-cofactorial automorphism-invariant subgroup|FULL LIST, MORE INFO |