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Let $G$ be a group which contains a linear subgroup of finite index. Is $G$ necessarily linear? What if $G$ is finitely generated?

I cannot seem to find anything which talks about "virtually linear" groups, and so I would hypothesise that "virtually linear $\Rightarrow$ linear". Also, many properties of linear groups are virtual properties, such as the Tits alternative and residually finite. However, these facts do not constitute a proof!

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What definition of linear groups do you use? Do you want to know if a finite extension of a linear group can be embedded into some general linear group? –  Vittorio Patriarca Jun 4 '12 at 15:20
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If by "linear" you mean can be embedded in a general linear group over some field $F$, then yes, virtually linear implies linear. If $|G:H|=r$ and $H$ embeds in ${\rm GL}(n,F)$, then the induced representation is an embedding of $G$ into ${\rm GL}(rn,F)$. –  Derek Holt Jun 4 '12 at 15:27
    
@DerekHolt: Thanks, that's what I was looking for! –  user1729 Jun 4 '12 at 15:42

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