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?


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).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.