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A group $G$ is locally solvable if all finitely generated subgroups are solvable.

A group $G$ is virtually locally solvable if it has a locally solvable subgroup of the finite index.

My question is:

$N\vartriangleleft G$ such that $G/N$ virtually locally solvable $\Rightarrow G$ virtually locally solvable?

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    I think that if $N=\left\langle T;T\trianglelefteq G,T\text{ locally solvable }\right\rangle $ is true (supposing that $G/N$ virtually locally solvable)2012-05-25

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