# Intermediately normal-to-characteristic implies intermediately characteristic in nilpotent

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties, when the big group is a nilpotent group. That is, it states that in a Nilpotent group (?), every subgroup satisfying the first subgroup property (i.e., Intermediately normal-to-characteristic subgroup (?)) must also satisfy the second subgroup property (i.e., Intermediately characteristic subgroup (?)). In other words, every intermediately normal-to-characteristic subgroup of nilpotent group is a intermediately characteristic subgroup of nilpotent group.

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## Contents

## Statement

In a nilpotent group, any intermediately normal-to-characteristic subgroup is an intermediately characteristic subgroup.

## Definitions used

### Intermediately normal-to-characteristic subgroup

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

A subgroup of a group is termed intermediately normal-to-characteristic in if whenever , with a normal subgroup of , is a characteristic subgroup of .

### Intermediately characteristic subgroup

`Further information: Intermediately characteristic subgroup`

A subgroup of a group is termed intermediately characteristic in if whenever , is a characteristic subgroup of .

### Intermediately subnormal-to-normal subgroup

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

A subgroup of a group is termed intermediately subnormal-to-normal in if whenever , with a subnormal subgroup of , is a normal subgroup of .

## Related facts

### Corollaries

## Facts used

- Intermediately normal-to-characteristic implies intermediately subnormal-to-normal
- Nilpotence is subgroup-closed
- Nilpotent implies every subgroup is subnormal

## Proof

**Given**: A nilpotent group , an intermediately normal-to-characteristic subgroup of .

**To prove**: Whenever is an intermediate subgroup, i.e. , is characteristic in .

**Proof**: By fact (2), is nilpotent and by fact (3), is a subnormal subgroup of .

By fact (1), is intermediately subnormal-to-normal in , and combinig this with the fact that is subnormal in yields that is normal in . Combining this with the given fact that is intermediately normal-to-characteristic in yields that is characteristic in , completing the proof.