Normal-isomorph-free subgroup
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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]
Definition
A subgroup of a group is termed normal-isomorph-free if it is a normal subgroup, and there is no other normal subgroup of the whole group isomorphic to it.
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 | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Normal-isomorph-containing subgroup | contains any isomorphic normal subgroup of whole group | |FULL LIST, MORE INFO | ||
| Characteristic-isomorph-free subgroup | characteristic and no isomorphic characteristic subgroup | |FULL LIST, MORE INFO | ||
| Characteristic subgroup | invariant under all automorphisms | Characteristic-isomorph-free subgroup, Series-isomorph-free subgroup|FULL LIST, MORE INFO | ||
| Series-isomorph-free subgroup | no other normal subgroup that is isomorphic and has isomorphic quotient | |FULL LIST, MORE INFO | ||
| Normal-isomorph-automorphic subgroup | normal and any isomorphic normal subgroup is automorphic | |FULL LIST, MORE INFO |
