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

From Groupprops
Jump to: navigation, search

Statement

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

Related facts

Facts used

  1. Penultimate term of lower central series is Abelian in nilpotent group of class at least three

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).