Lie Group structure on $S^2$

Question

I see that people say there is no Lie group structure on $S^2$. But $S^2$ can be identified with $SU(2)/U(1)$ by the Hopf fiberation. Since $U(1)$ is also a normal subgroup in $SU(2)$, can't you associate the quotient group structure to $S^2$?

What is wrong with this association? Thanks!

Answer 1

The group $U(1)$ is not a normal subgroup of $SU(2)$. For instance,\begin{multline}\begin{bmatrix}\frac12+\frac i2&-\frac12+\frac i2\\\frac12+\frac i2&\frac12-\frac i2\end{bmatrix}.\begin{bmatrix}\cos(\theta)&-\sin(\theta)\\\sin(\theta)&\cos(\theta)\end{bmatrix}.\begin{bmatrix}\frac12+\frac i2&-\frac12+\frac i2\\\frac12+\frac i2&\frac12-\frac i2\end{bmatrix}^{-1}=\\=\begin{bmatrix}\cos(\theta)+i\sin(\theta)&0\\0&\cos(\theta)-i\sin(\theta)\end{bmatrix}\notin U(1).\end{multline}

And people don't just say that there is no Lie group structure on $S^2$. They prove it.