In this paper, we answer affirmatively the problem proposed by A. Selvitella in his paper "Nondegenracy of the ground state for quasilinear Schr\"odinger Equations" (see Calc. Var. Partial Differ. Equ., {\bf 53} (2015), pp 349-364): every ground state of equation \begin{eqnarray*}-\Delta u-u\Delta |u|^2+\omega u-|u|^{p-1}u=0&&\text{in }\mathbb{R}^N\end{eqnarray*} is nondegenerate for $10$ is a given constant and $N\ge1$. We also derive further properties on the linear operator associated to ground states of above equation.
