  We show the uniqueness of strictly convex closed smooth self-similar
solutions to the $\alpha$-Gauss curvature flow with $(1/n) < \alpha < 1+(1/n)$.
We introduce a Pogorelov type computation, and then we apply the strong maximum
principle. Our work combined with earlier works on the Gauss Curvature flow
imply that the $\alpha$-Gauss curvature flow with $(1/n) < \alpha < 1+(1/n)$
shrinks a strictly convex closed smooth hypersurface to a round sphere.
