= Finite groups cannot act freely on Euclidean space
If a nontrivial finite group acted freely on $\mathbb R^n$, a prime-order cyclic subgroup $C_p$ would too. The quotient is an $n$-manifold with contractible universal cover. The singular version of <cellular free resolution from a contractible universal cover> identifies its cohomology with <group cohomology> of $C_p$. The <periodic resolution of a finite cyclic group> gives $H^{2j}(C_p;\mathbb Z)=\mathbb Z/p$ for every $j\geq1$. But <Poincaré duality for noncompact manifolds> makes the quotient cohomology zero above dimension $n$. This contradiction excludes the action, for arbitrary homeomorphisms, not only Euclidean isometries.
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