I would like to know how I can prove that $SU(2)$ acts transitively on $S^{3}$. Currently, I want to show that $SU(2)$ is a group of isometries of $S^{3}$.
2026-02-23 13:34:46.1771853686
Isometry group of $3$-sphere
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Consider $S^3 = \{(x,y)^t\in\mathbb C^2|\|(x,y)\|=1\}\subset \mathbb C^2$. Then the matrix $A:=\left(\begin{matrix}x&-\overline{y}\\y&\overline x\end{matrix}\right)$ is unitary, has determinant $1$ and $A\cdot(1,0)^t = (x,y)^t$.