I have been reading through a book of Robinson where it is mentioned (informally!) that solvable groups have "many" subnormal subgroups (subgroups $H<G$ with $H=H\_0 \lhd H\_1 \lhd \ldots \lhd H\_n = G$). There seems to be quite a body of work on this, but it brought me to ponder on:


$\textbf{Question:}$ Given an infinite solvable group $G$, are there "few" malnormal subgroups (i.e. $H \cap gHg^{-1} = e$ for any $g \in G \setminus H$)?


I'm open to any interpretation of "few" (from "none" to "if there is one then it must be [insert property]" to ...) or any example.


[Edit: I forgot to put "finitely generated" in the question.]



Malnormal subgroups in solvable groups
In general, if $G=N \rtimes H$ is a semidirect product in which all nontrivial elements of $H$ act fixed-point-freely on $N$, then $H$ is malnormal in $G$.


For example, if $K$ is any group and $H$ is any torsion-free group, then $H$ is malnormal in the restricted wreath product $K \wr H$.


If we choose $K$ and $H$ to be finitely generated and solvable then $G$ will be too. We could take $G = {\mathbb Z} \wr {\mathbb Z}$, for example.


