Pointwise inner automorphisms in a free nilpotent group

Abstract

An automorphism θ of a group G is pointwise inner if θ (t) is conjugate to t for any t ∈ G. It is known that in a free group, each pointwise inner automorphism is inner. We prove that this property remains true in a free nilpotent group. In fact, more generally, an automorphism of a non‐cyclic free group fixing each normal subgroup is inner. We show that in a non‐abelian free nilpotent group of class c, this last property is true if c = 2, but false if c > 2.