= Free actions on even-dimensional spheres have order at most two
For every nonidentity element of a <free group action> on $S^{2r}$, its action map has no fixed point and hence degree minus one by <fixed-point-free sphere maps are homotopic to the antipodal map>. Degree of homeomorphisms defines a homomorphism into $\{1,-1\}$, whose kernel is trivial, since no nonidentity action map can have degree one. Thus the acting group has order at most two. For $S^0$, freeness on its two points gives the same conclusion directly.
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