Exercise 15, § 4

Let H be a connected closed subgroup of G. Then H is said to be clean if it is not contained in any connected closed subgroup of maximal rank distinct from G.

a) Show that H is clean if and only if its centralizer in G is equal to C(G). In particular, if H is clean then $C(H) = C(G)\cap H$.

b) Assume from now on that H is clean. Show that, for every maximal torus S of H, we have $C(H) = S\cap C(G)$.

c) Let $H'$ be a connected closed subgroup of G containing H. Prove that rk $H<$ rk $H'$, and deduce that H is clean in $H'$.

d) Let K be a connected closed subgroup of H, clean in H and containing a regular element of G. Show that K is clean in G.