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 |