# Non-elementary examples of nearly normal subgroups

$H$ is called a nearly normal subgroup of a group $G$ if it is of finite index in its normal closure, $H^G:=\cup_{g\in G} gHg^{-1}$, in $G$. Clearly, every normal subgroup or subgroup of finite index of $G$ is nearly normal. I am not interested in finite subgroups either. I am looking for infinite nearly normal subgroups of infinite index in well-known groups. By a well-known group, I mean a group that some of its basic properties like amenability, growth or its geometric nature (the spaces it acts) have been studied in the literature.

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@ Will: Thank you for your answer. It is stimulating. But I am looking for very concrete examples. For instance, is $SL(2,\mathbb{Z})$ nearly normal in $GL(2,\mathbb{Q})$? –  Vahid Shirbisheh May 22 '12 at 17:27