Is there a classification for infinite finitely generated solvable groups all of whose abelian normal subgroups are finite?


I mean by classification something like presentation.


Edited: Is there an infinite finitely generated solvable group $G$ all of whose abelian normal subgroups are finite and $G$ is not residually finite?


Thanks in advance for any help.



Finitely generated solvable groups all of whose abelian normal subgroups are finite
To your edited question:



> 
> There is an infinite finitely generated solvable group with no infinite normal abelian subgroup.
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The example is an extension of $\mathbf{Z}/p\mathbf{Z}$ by the lamplighter $(\mathbf{Z}/p\mathbf{Z})^2\wr\mathbf{Z}$.


Fix a prime $p$ congruent to -1 modulo 4 (so that -1 has no square root mod $p$). Consider a group $H\_p$ made up of formal sums $\sum a\_ne\_n\oplus\sum b\_nf\_n\oplus c$, where $a\_n,b\_n,c\in\mathbf{F}\_p$ (all but finitely being 0), and $e\_n,f\_n$ is a fixed "basis". Define the "Heisenberg-like product" $(\sum a\_ne\_n\oplus\sum b\_nf\_n\oplus c)(\sum a'\_ne\_n\oplus\sum b'\_nf\_n\oplus c')=\sum a''\_ne\_n\oplus\sum b''\_nf\_n\oplus c''$, where $a''\_n=a\_n+a'\_n$, $b''\_n=b\_n+b'\_n$, $c''=c+c'+\sum a\_nb'\_n-a'\_nb\_n$. This is a group (it is the quotient of an infinite direct sum of copies of the Heisenberg by a central subgroup, at least if $p\neq 2$). 


We denote by $z$ the element $0\oplus 0\oplus 1$ of $H\_p$. 
There is an obvious automorphism $\alpha$ of $H\_p$ shifting the indices, namely mapping $e\_n\mapsto e\_{n+1}$, $f\_n\mapsto f\_{n+1}$, $z\mapsto z$. Let us also consider the involution $s: e\_n\mapsto f\_n\mapsto e\_n$, $z\mapsto -z$, which commutes with $\alpha$.


Consider the semidirect product $G\_p=H\_p\rtimes(\mathbf{Z}\times\mathbf{Z}/2)$, where $\mathbf{Z}\times\mathbf{Z}/2$ is identified to $\langle\alpha,s\rangle$. Then $G$ is 3-step solvable, and finitely generated (by 3 generators, say $\alpha$, $s$, and $e\_0$).


I claim that $G\_p$ has no infinite abelian normal subgroup, and more precisely that every abelian normal subgroup $N$ of $G\_p$ is contained in the normal cyclic subgroup $\langle z\rangle$ of order $p$. Otherwise, the projection $N'$ of $N$ on $G\_p$ is a nontrivial abelian normal subgroup. Since $H'\_p=H\_p/Z$ is its own centralizer in $G'\_p=G\_p/Z$, we have $N'\cap H'\_p$ nontrivial. So we can suppose that $N\subset H\_p$. Thus $N'$ is an $\langle\alpha,s\rangle$-invariant subgroup of $H'\_p$, i.e. a $\mathbf{F}\_{p^2}[\alpha^{\pm 1}]$-submodule of $H'\_p$. Here we defined this module structure by letting a given square root of -1 (which is in $\mathbf{F}\_{p^2}\smallsetminus\mathbf{F}\_{p}$) act as $s$. Since $N'$ is nonzero, it contains some element of the form $w=\lambda\_0e\_0+\lambda\_1e\_1+\dots \lambda\_ke\_k$, where $k\ge 0$, $\lambda\_i\in\mathbf{F}\_{p^2}$, and $\lambda\_0\lambda\_k\neq 0$. So it also contains the element $w'=\lambda\_0e\_{-k}+\dots \lambda\_ke\_0$. But any two lifts in $N$ of $w$ and $w'$ do not commute and we get a contradiction.




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Addendum to give details of my comments to the question:



> 
> Every infinite solvable group has an infinite 2-step nilpotent characteristic subgroup with finite derived subgroup. In particular, every infinite solvable group $G$ that is residually finite has an infinite abelian normal subgroup (which can be chosen characteristic if $G$ is finitely generated).
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(A group $G$ is by definition 2-step nilpotent if its derived subgroup $G'$ is central; this includes abelian groups.)


The second assertion follows from the first: choose $N$ as in the first assertion; by residual finiteness, there is a finite index normal subgroup $M$ such that $N'\cap M=\{1\}$; if $G$ is residually finite then $M$ can be chosen to be characteristic. Hence $N\cap M$ is an infinite abelian normal subgroup, and is characteristic if $M$ is characteristic.


For the first assertion, let $N$ be the last infinite term in the derived series. So $N'$ is finite. Hence, the centralizer of $N'$ in $N$ has finite index in $N$, is an infinite characteristic subgroup with central finite derived subgroup, that is, 2-step nilpotent.


