  We prove that positive solutions $u\in H^s(\mathbb{R}^N)$ to the equation
$(-\Delta )^s u+ u=u^p$ in $\mathbb{R}^N$ are nonradially nondegenerate, for
all $s\in (0,1)$, $N\geq 1$ and $p>1$ strictly smaller than the critical
Sobolev exponent. By this we mean that the linearized equation $(-\Delta )^s w+
w-pu^{p-1}w = 0$ does not admit nonradial solutions beside the directional
derivatives of $u$. Letting $B$ be the unit centered ball and $\lambda_1(B)$
the first Dirichlet eigenvalue of the fractional Laplacian $(-\Delta )^s$, we
also prove that positive solutions to $(-\Delta )^s u+\lambda u=u^p$ in ${B}$
with $u=0$ on $\mathbb{R}^N\setminus B$, are nonradially nondegenerate for any
$\lambda> -\lambda_1(B)$ in the sense that the linearized equation does not
admit nonradial solutions. From the latter result, we then deduce uniqueness
and full nondegeneracy of positive solutions in the special case $N=1$ and
$\lambda = 0$. Moreover, we deduce uniqueness of positive ground state
solutions in the case $N=1$, $\lambda \ge 0$.
