Restriction of a map to a Maximal Torus is surjective

Question

Let $f:G\to H$ be a surjective homomorphism from a compact Lie group $G$ to an Abelian Lie group $H$. Then, is the restriction of the map $f$ to the maximal torus $T$ of $G$ is also surjective ?

Answer 1

Assume that $G$ is connected. Then every element $g\in G$ is conjugate to some element of $T$. Hence $g = xtx^{-1}$ for $t\in T$ and $x\in G$. Note that

$$f(g) = f(xtx^{-1}) = f(x)f(t)f(x)^{-1} = f(t)$$

because $H$ is abelian. This implies that $f(T) = f(G) = H$ and therefore, $f_{\mid T}$ is surjective.

If $G$ is not connected, this is not true by obvious reasons. For example pick $$G= H = \mathbb{S}^1\times \mathbb{Z}/2\mathbb{Z}$$ and $$f = 1_{\mathbb{S}^1\times \mathbb{Z}/2\mathbb{Z}}$$ Then $T = \mathbb{S}^1\times \{0\}$ and $f(T)$ is not $H$.