# Subgroup whose normalizer equals its normal closure

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This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: tightly 2-subnormal subgroup and 2-hypernormalized subgroup

View other subgroup property conjunctions | view all subgroup properties

## Definition

A subgroup of a group is termed a **subgroup whose normalizer equals its normal closure** if it satisfies the following equivalent conditions:

- The normalizer of in equals the normal closure of in .
- It is both a 2-hypernormalized subgroup (its normalizer is normal) and a subgroup with unique 2-subnormal series (it has a unique subnormal series of length two).

## Relation with other properties

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

2-hypernormalized subgroup | normalizer is normal | |FULL LIST, MORE INFO | ||

Subgroup with unique 2-subnormal series | has unique subnormal series of length two; equivalently, its normal closure is the normal core of its normalizer | |FULL LIST, MORE INFO | ||

2-subnormal subgroup | normal subgroup of normal subgroup | |FULL LIST, MORE INFO |