Exercise 8, § 2

Let $G$ be a quasi-topological group (Exercise 5).

a) Extend Proposition 8 to $G$ (show first that $xKx^{-1} \subset K$ if $x \in H$). Give an example of semi-topological group which is not quasi-topological and for which Proposition 8 is not valid [cf. § 1, Exercise 2 b)].

b) Let $H, K$ be two subgroups of $G$ such that $H \triangleright K$ and $K$ contains the commutator subgroup of $H$. Show that $K$ contains the commutator subgroup of $\overline{H}$ (analogous method).

c) Deduce from b) that if every subset of $G$ consisting of a single point is closed, then the closure of every solvable (resp. commutative) subgroup in $G$ is solvable (resp. commutative) (argue on the length of the derived series of the subgroup under consideration).