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

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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


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).