Exercise 1, § 2

Let G be a connected compact Lie group, $g$ an element of G.

a) Show that there exists an integer $n\geq 1$ such that the centralizer $Z(g^n)$ is connected (prove that, for suitable $n$, the closed subgroup generated by $g^n$ is a torus).

b) Assume that the dimension of Ker(Ad $g^n-1$) is independent of $n(n\geq 1)$. Prove that $Z(g)$ is connected.

c) If $g^n$ is regular for all $n\geq 1, Z(g)$ is connected.