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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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    You must make some assumptions on $N$, otherwise it is clearly false.2012-05-24
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    is it clearly false? Why?2012-05-24
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    Take $N$ to be some ugly group and consider $G\cong G/N\times N$.2012-05-24
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    It is clearly flase because you can take $N=G$ for any $G$; then $G/N$ is certainly virtually locally solvable: it's even solvable! But there is no reason why this would imply that $G$ is virtually locally solvable.2012-05-24
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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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