Show is a solvable group.
The group operation is matrix multiplication.
This is not the usual definition, and it's a rather misleading and even incorrect one (imo, of course. See following note), but it never minds: since the group is finitely generated then is finitely generated and thus can be decomposed in a direct product of cyclic groups...
Note: for your definition to work it MUST be that G is fin. generated, or at least that all its abelian factor groups are, otherwise it fails: an infinitely generated abelian group wouldn't be solvable according to your definition which, of course, is absurd.