Homomorphism image of maximal torus

Question

I’m asked to show given a surjective homomorphism between compact connected Lie groups: $G$ and $H$, then maximal torus in $G$ are mapped to maximal torus in $H$. Well, I’m not sure how can I use the subjectivity to show that the torus we get by homomorphism is actually maximal. I doubt the maximal torus theorem comes into play, but I don’t see how, any suggestions?

Answer 1

Hints:

1. Let $S$ be a compact connected Lie group, $K$ being a normal closed subgroup of $S$. If the factor group $S/K$ is an Abelian group, then there exists a connected closed Abelian subgroup $R$ of $S$ such that $S=KR$ and $|R\cap K|<\infty$.

2. Let $G$ and $H$ be compact connected groups Lie, $T$ being a maximal torus of $G$, $f:G\rightarrow H$ being a surjective homomorphism. Since $f(T)$ is a connected Abelian subgroups of $H$, there exists a maximal torus $T'$ of $H$ such that $f(T)<T'$. Let $S=f^{-1}(T')$ and $K=Ker(f)$. Then $S/K\cong T'$ is an Abelian group.