# Nilpotent and every abelian characteristic subgroup is central implies class at most two

From Groupprops

## Contents

## Statement

If is a Nilpotent group (?) and satisfies the property that every Abelian characteristic subgroup of it is central, then the nilpotence class of is at most two.

## Related facts

## Facts used

## Proof

The proof follows direct from fact (1): it shows that in a nilpotent group of class at least three, we have an Abelian characteristic subgroup that is not central (namely, the penultimate term of the lower central series).