The dimension of spheres with smooth one fixed point actions

Bak A, Morimoto M. The dimension of spheres with smooth one fixed point actions. Forum Mathematicum. 2005;17(2):199-216.The article proves that there are smooth one fixed point actions of the alternating group of degree 5 on the 8-dimensional sphere. It follows that a sphere has a smooth one fixed point action of some finite group if and only if the dimension of the sphere is greater than or equal to 6