# Normalizer-relatively normal subgroup

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This article describes a property that can be evaluated for a triple of a group, a subgroup of the group, and a subgroup of that subgroup.

View other such properties

## Definition

Suppose are groups. We say that is a **normalizer-relatively normal subgroup** of with respect to if the following equivalent conditions are satisfied:

- Whenever is such that is normal in , is also normal in .
- The normalizer is contained in the normalizer .
- is normal in .

## Relation with other properties

### Stronger properties

- Middle-characteristic subgroup
- Strongly closed subgroup
- Weakly closed subgroup:
`For full proof, refer: Weakly closed implies normalizer-relatively normal`