Finitely presented and solvable not implies polycyclic
This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., finitely presented solvable group) need not satisfy the second group property (i.e., polycyclic group)
View a complete list of group property non-implications | View a complete list of group property implications
Get more facts about finitely presented solvable group|Get more facts about polycyclic group
It is possible to have a finitely presented solvable group -- a group that is both a finitely presented group and a solvable group -- such that is not a polycyclic group. In particular, this means that is not a Noetherian group, i.e., it has a subgroup that is not finitely generated.
Further information: Baumslag-Solitar group:BS(1,2)
The group , the Baumslag-Solitar group with parameters , works:
The normal closure of in this group, which is also the derived subgroup of , is isomorphic to the group of 2-adic rationals, i.e., the group of all rationals with denominators powers of 2.