Exercise 11, § 2

Let G be a compact Lie group. A subgroup S of G is said to be of type (C) if it is equal to the closure of a cyclic subgroup that is of finite index in its normalizer.

a) Let S be a subgroup of type (C); show that $S_0$ is a maximal torus of $G_0$, and that S is the direct product of $S_0$ and a finite cyclic subgroup.

b) Prove that every element $g$ of G is contained in a subgroup of type (C) (consider the group generated by $g$ and a maximal torus of $Z(g)_0$).

$^{11}$For a proof of this theorem, see for example S. LEFSCHETZ, Intersections and

transformations of complexes and manifolds, Trans. Amer. Math. Soc. 28 (1926),

pp. 1-49. c) Let S be a subgroup of type (C), and $s$ an element of S whose set of powers is dense in S. Show that every element of $sG_0$ is conjugate under Int(G$_0$) to an element of $sS_0$ (use the method of Exerc. 10).

d) Denote by $p: G\rightarrow G/G_0$ the quotient map. Show that the map $S \rightarrow p(S)$ induces a bijection from the set of conjugacy classes of subgroups of G of type (C) to the set of conjugacy classes of cyclic subgroups of $G/G_0$.

e) Let S be a subgroup of G of type (C); denote by $S_{\rho}$ the set of elements of S whose class in $S/S_0$ generates $S/S_0$. Show that two elements of $S_{\rho}$ that are conjugate in G are conjugate in $N_G(S)$.