# Commutator of a normal subgroup and a subset implies 2-subnormal

From Groupprops

This article describes a computation relating the result of the commutator operator on two known subgroup properties or properties of subsets of groups: (i.e., normal subgroup and subset of a group), to another known subgroup property (i.e., 2-subnormal subgroup)

View a complete list of commutator computations

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., subgroup realizable as the commutator of a normal subgroup and a subset) must also satisfy the second subgroup property (i.e., 2-subnormal subgroup)

View all subgroup property implications | View all subgroup property non-implications

Get more facts about subgroup realizable as the commutator of a normal subgroup and a subset|Get more facts about 2-subnormal subgroup

## Contents

## Statement

Suppose is a group, is a normal subgroup of , and is any subset of . Then, the subgroup:

is a 2-subnormal subgroup of .

## Related facts

### Commutators between things of the same type

- Normality is commutator-closed: The commutator of two normal subgroups is normal.
- Characteristicity is commutator-closed
- Commutator of subnormal subgroups is subnormal iff their join is subnormal

### Similar facts

- Commutator of a group and a subset implies normal
- Commutator of a 2-subnormal subgroup and a subset implies 3-subnormal
- Commutator of two subgroups is normal in join
- Commutator of a 3-subnormal subgroup and a finite subset implies subnormal

### Opposite facts

- Commutator of a normal subgroup and a subgroup not implies normal
- Commutator of a 3-subnormal subgroup and a subset not implies subnormal
- Commutator of a 3-subnormal subgroup and a finite subset not implies 4-subnormal

## Facts used

- Subgroup normalizes its commutator with any subset: If and is a subset of , normalizes .

## Proof

**Given**: A group , a normal subgroup of , a subset of .

**To prove**: is 2-subnormal in .

**Proof**:

- (
**Given data used**: is normal in ): Since is normal in , . Thus, . - (
**Fact used**: fact (1)): By fact (1), normalizes . Combining this with step (1), we get that is normal in . - We thus have that is normal in , and is normal in . Thus, is 2-subnormal in .