# Normal-isomorph-free subgroup

From Groupprops

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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 |