# Sub-cofactorial automorphism-invariant subgroup

## Definition

A subgroup $H$ of a group $G$ is termed a sub-cofactorial automorphism-invariant subgroup if there exists an ascending chain of subgroups: $H = H_0 \le H_1 \le \dots \le H_n = G$

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

## 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