# Intermediately normal-to-characteristic implies intermediately subnormal-to-normal

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., intermediately normal-to-characteristic subgroup) must also satisfy the second subgroup property (i.e., intermediately subnormal-to-normal subgroup)

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This fact is an application of the following pivotal fact/result/idea:characteristic of normal implies normal

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## Contents

## Statement

### Property-theoretic statement

The subgroup property of being intermediately normal-to-characteristic implies the subgroup property of being intermediately subnormal-to-normal.

## Definitions used

### Intermediately subnormal-to-normal subgroup

`Further information: Intermediately subnormal-to-normal subgroup`

A subgroup of a group is termed **intermediately subnormal-to-normal** if whenever is a subgroup containing such that is 2-subnormal in , then is normal in .

### Intermediately normal-to-characteristic subgroup

`Further information: Intermediately normal-to-characteristic subgroup`

A subgroup of a group is termed **intermediately normal-to-characteristic** if whenever is a subgroup containing such that is normal in , is characteristic in .

## Facts used

## Proof

**Given**: A subgroup of a group , such that is intermediately normal-to-characteristic in .

**To prove**: For any subgroup of containing , such that is 2-subnormal in , is normal in .

**Proof**: Since is 2-subnormal in , there exists a subgroup such that .

Now, is a subgroup of , and is normal in . Since is intermediately normal-to-characteristic in , is characteristic in .

Thus, is characteristic in , that in turn is normal in . By fact (1), is normal in , completing the proof.