TITLE: Radical Locally Solvable?
QUESTION [6 upvotes]: A group $G$ is locally solvable if all finitely generated subgroups are
solvable.
A group $G$ is locally finite if all finitely generated subgroups are
finite.
A group $G$ is virtually locally solvable if it has a locally solvable
subgroup of the finite index.
Let be $R(S)=\left\langle T\,;\,T\trianglelefteq G\,,\,T\text{ locally solvable }\right\rangle $
My question are:
1)Is $\,R(S)\,$ locally solvable?
2) If 1) is true: $G$ locally finite, $R(S)$ locally solvable and $G/R(S)$ virtually locally solvable $\Rightarrow G$
virtually locally solvable?

REPLY [3 votes]: No, your group $R(S)$ need not be locally solvable.  A counterexample due to P. Hall is described in Part 2 of Robinson's "Finiteness Conditions and Generalized Soluble Groups".  See the development leading up to, and the proof of, Theorem 8.19.1 on page 91 (and the corollary).