Exercise 1, Appendix 1

Let G be a connected compact group. Denote by $d(G)$ the upper bound of the dimensions of the quotients of G that are Lie groups. Assume that $d(G)<\infty$.

a) Let K be a closed normal subgroup of G; show that $d(G/K)\leq d$(G), and that $d(G/K) =d(G)$ if K is totally discontinuous. b) Show that D(G) is a Lie group, and that the kernel of the homomorphism $(x, y) \rightarrow xy$ from $C(G)_0\times D(G)$ to G is finite.

c) Let $p=d(C(G)_0)$. Then $p <\infty$; prove that there exists a totally discontinuous compact group D and a homomorphism $i:\mathbf{Z}^p\rightarrow D$ with dense image, such that $C(G)_0$ is isomorphic to $(\mathbf{R}^p\times D)/\Gamma$, where $\Gamma$ is the image of $\mathbf{Z}^p$ under the homomorphism $x \rightarrow (x, i(x))$ (write $C(G)_0$ as a projective limit of tori of dimension $p$).

d) Assume that G is locally connected; show that G is then a Lie group.