Past exam of the lie Groups and Differentiable Manifolds course of the University of Oxford / 2008 / A / 3 / b

A finite discrete group has trivial identity component, so its only torus is $T=\{1\}$. Thus $N_G(T)=G$ and, under the definition $N_G(T)/T$,The alternative quotient by the centralizer would give a different answer for a disconnected group; it is not the convention used here.

A maximal torus is $T=S{O}_2$, which is normal of index two in $O_2$. ThereforeA reflection represents its nontrivial element and conjugates a rotation of angle $\theta$ to the rotation of angle $-\theta$.

Use the given torus of independent rotations in $n$ mutually orthogonal two-planes and a fixed final axis. A normalizer element permutes the two-plane rotation characters, and on each plane either preserves or reverses its orientation. Its induced action therefore permutes the $n$ angle coordinates and changes their signs independently.

Every such signed permutation is realized. Permutations of two-dimensional blocks have determinant one. A reflection in any chosen plane reverses that angle; reflecting the remaining one-dimensional axis as necessary restores total determinant one. The kernel consists precisely of independent plane rotations: commuting with a generic torus element forces preservation of its distinct rotation planes, and the determinant condition then fixes the final axis. Thus the kernel is $T$ andThis is the signed permutation group, the Weyl group of type $B_n$.