# Characteristic-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 **characteristic-isomorph-free** if it is a characteristic subgroup and there is no characteristic subgroup of the whole group isomorphic to it.

## Relation with other properties

### Stronger properties

- Isomorph-free subgroup
- Normal-isomorph-free subgroup:
*For proof of the implication, refer Normal-isomorph-free implies characteristic-isomorph-free and for proof of its strictness (i.e. the reverse implication being false) refer Characteristic-isomorph-free not implies normal-isomorph-free*.

### Weaker properties

- Characteristic subgroup:
*For proof of the implication, refer Characteristic-isomorph-free implies characteristic and for proof of its strictness (i.e. the reverse implication being false) refer Characteristic not implies characteristic-isomorph-free*.