  Recently a new class of scalarized black holes in Einstein-Gauss-Bonnet (EGB)
theories was discovered. What is special for these black hole solutions is that
the scalarization is not due to the presence of matter, but {it is induced} by
the curvature of spacetime itself. Moreover, more than one branch of scalarized
solutions can bifurcate from the Schwarzschild branch, and these scalarized
branches are characterized by the number of nodes of the scalar field. The next
step is to consider the linear stability of these solutions, which is
particularly important due to the fact that the Schwarzschild black holes lose
stability at the first point of bifurcation. Therefore we here study in detail
the radial perturbations of the scalarized EGB black holes. The results show
that all branches with a nontrivial scalar field with one or more nodes are
unstable. The stability of the solutions on the fundamental branch, whose
scalar field has no radial nodes, depends on the particular choice of the
coupling function between the scalar field and the Gauss-Bonnet invariant. We
consider two particular cases based on the previous studies of the background
solutions. If this coupling has the form used in \cite{Doneva:2017bvd} the
fundamental branch of solutions is stable, except for very small masses. In the
case of a coupling function quadratic in the scalar field \cite{Silva:2017uqg},
though, the whole fundamental branch is unstable.
