$H$ 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 wellknown groups. By a wellknown 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.
1 Answer
Take any infinite group with an infiniteindex normal subgroup. Say, the free group on two generators with the commutator subgroup. Then take a finite index subgroup of the normal subgroup. This particular normal subgroup is the free group on countably infinite generators, and so has many finiteindex subgroups. Most are not normal, but all are nearly normal.
This feels somewhat unsatisfying, though. Presumably the only way to get something satisfying is to find a nearly normal subgroup whose definition is more natural than the corresponding normal subgroup. I do not know any subgroups like that.

$\begingroup$ @ 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 $GL(2,\mathbb{Q})$? $\endgroup$– user23860May 22, 2012 at 17:27

$\begingroup$ @Vahid: SL(2, Q) is simple modulo its center, so SL(2,Z) is not nearly normal in GL(2,Q). If you want concrete examples, take any finite index nonnormal subgroup H<G_1. Then H is nearly normal in G=G_1\times G_1. For instance, take G_1=SL(2,Z) and H<G_1 be the subgroup of matrices which reduce to uppertriangular modulo some prime p. $\endgroup$– MishaMay 22, 2012 at 17:50