Retraction-invariant characteristic subgroup
From Groupprops
This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: retraction-invariant subgroup and characteristic subgroup
View other subgroup property conjunctions | view all subgroup properties
Contents
Definition
A subgroup of a group is termed a retraction-invariant characteristic subgroup if it satisfies the following two conditions:
- It is a retraction-invariant subgroup, i.e., any retraction of the whole group sends the subgroup to itself.
- It is a characteristic subgroup, i.e., any automorphism of the whole group sends the subgroup to itself.
Relation with other properties
Stronger properties
| property | quick description | proof of implication | proof of strictness (reverse implication failure) | intermediate notions |
|---|---|---|---|---|
| Fully invariant subgroup | invariant under all endomorphisms | |FULL LIST, MORE INFO |
Weaker properties
| property | quick description | proof of implication | proof of strictness (reverse implication failure) | intermediate notions |
|---|---|---|---|---|
| Characteristic subgroup | invariant under all automorphisms | |FULL LIST, MORE INFO | ||
| Normal subgroup | invariant under all inner automorphisms | Retraction-invariant normal subgroup|FULL LIST, MORE INFO | ||
| Retraction-invariant normal subgroup | normal and retraction-invariant | |FULL LIST, MORE INFO |
