Solution
= Solution
$SU(2)=\{A:A^*A=I,\det A=1\}$; every element is $\begin{pmatrix}a&b\\-\bar b&\bar a\end{pmatrix}$ with $|a|^2+|b|^2=1$, giving $S^3$. A maximal torus is $T=\{\operatorname{diag}(e^{it},e^{-it})\}$. A noncentral conjugacy class is determined by trace $2\cos t$, has stabilizer $T$, and is $SU(2)/T\simeq S^2$; it meets $T$ at the two inverse diagonal elements. Since $SU(2)$ has no nontrivial continuous characters, $\det\rho=1$.