Conjugate of a nilpotent subgroup

Question

Suppose that $G$ is a group with $H$ being a nilpotent subgroup of $G$. Let $g\in G$. Is is true that $gHg^{-1}$ is nilpotent in $G$?

Answer 1

The map $f:H\to gHg^{-1}$ defined by $f(h)=ghg^{-1}$ is an isomorphism. So $gHg^{-1}$ is isomorphic to $H$, and hence nilpotent.