spheres that are Lie groups

spheressatisfy

Theorem - The only spheres (http://planetmath.org/Sphere) that are Lie groups are $S^0$, $S^1$ and $S^3$.

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Proof: $$ It is known that $S^0$, $S^1$ and $S^3$ have a Lie group .

On the other , we have seen in the parent entry (http://planetmath.org/CohomologyOfCompactConnectedLieGroups) that the cohomology groups (http://planetmath.org/DeRhamCohomology) of a compact connected Lie group $G$ satisfy

$$H^1(G;ℝ)=0⟹H^3(G;ℝ)\ne 0$$

The result then follows from the fact that the of spheres satisfy $H^1(S^n;ℝ)=0$ and $H^3(S^n;ℝ)=0$ for $n\ne 1,3$. $\mathrm{\square }$