# 2-subnormal implies conjugate-permutable

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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., 2-subnormal subgroup) must also satisfy the second subgroup property (i.e., conjugate-permutable subgroup)

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## Contents

## Statement

### Verbal statement

Any 2-subnormal subgroup (i.e. a subgroup that is normal inside its normal closure) is conjugate-permutable.

### Symbolic statement

Suppose is a 2-subnormal subgroup of . In other words, is a normal subgroup inside its normal closure in . Then, for any , and are permuting subgroups.

### Property-theoretic statement

The subgroup property of being 2-subnormal is stronger than the subgroup property of being conjugate-permutable.

## Definitions used

### 2-subnormal subgroup

`Further information: 2-subnormal subgroup`

### Conjugate-permutable subgroup

`Further information: Conjugate-permutable subgroup`

## Facts used

## Proof

### Hands-on proof

### Proof modulo the fact on automorph-permutable subgroups

A 2-subnormal subgroup is an normal subgroup of a normal subgroup. Since any normal subgroup is automorph-permutable, a 2-subnormal subgroup is an automorph-permutable subgroup of a normal subgroup. Using the above fact, we conclude that any 2-subnormal subgroup is conjugate-permutable.