# Normal-homomorph-containing 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-homomorph-containing** if is a normal subgroup of , and for any homomorphism such that is also a normal subgroup of , we have .

## Relation with other properties

### Stronger properties

### Weaker properties

- Weakly normal-homomorph-containing subgroup
- Strictly characteristic subgroup:
*For proof of the implication, refer Normal-homomorph-containing implies strictly characteristic and for proof of its strictness (i.e. the reverse implication being false) refer Strictly characteristic not implies normal-homomorph-containing*. Also related: - Normal-isomorph-containing subgroup