# Nilpotent implies every normal subgroup is potentially characteristic

From Groupprops

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., Normal subgroup (?)) must also satisfy the second subgroup property (i.e., Potentially characteristic subgroup (?)). In other words, every normal subgroup of nilpotent group is a potentially characteristic subgroup of nilpotent group.

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This fact is related to: NPC conjecture

View other facts related to NPC conjectureView terms related to NPC conjecture |

## Contents

## Statement

In a nilpotent group, every normal subgroup is a potentially characteristic subgroup: it can be realized as a characteristic subgroup inside a possibly bigger group.

## Related facts

- Finite normal implies potentially characteristic
- Central implies potentially characteristic
- Finite implies every normal subgroup is potentially characteristic
- Abelian implies every subgroup is potentially characteristic

## Facts used

## Proof

The proof follows from fact (1), and the observation that the hypercenter of a nilpotent group equals the whole group.