Isomorph-automorphic normal subgroup
From Groupprops
This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: isomorph-automorphic subgroup and normal subgroup
View other subgroup property conjunctions | view all subgroup properties
Contents
Definition
A subgroup of a group is termed an isomorph-automorphic normal subgroup if it satisfies the following two conditions:
- It is a normal subgroup.
- Every subgroup of the whole group isomorphic to it is an automorphic subgroup, i.e., there is an automorphism sending one subgroup to the other.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Isomorph-free subgroup | No other isomorphic subgroup | (see also list of examples) | ||
| Order-automorphic normal subgroup |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Isomorph-normal subgroup | every isomorphic subgroup of whole group is normal | isomorph-normal not implies isomorph-automorphic (see also list of examples) | ||
| Isomorph-automorphic subgroup | ||||
| Normal subgroup |
