Finitely generated and solvable not implies finitely presented

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 generated solvable group) need not satisfy the second group property (i.e., finitely presented group)
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Statement

There exists a finitely generated solvable group (i.e., a group that is both finitely generated and solvable) that is not finitely presented, i.e., it has no presentation with finitely many generators and finitely many relations.

Related facts

Similar facts

• Finitely generated not implies finitely presented
• Noetherian not implies finitely presented: A Noetherian group (also called snder group) is a group in which every subgroup is finitely generated. Such a group need not be finitely presented.

Opposite facts

• Polycyclic implies finitely presented: A polycyclic group can be defined as a Noetherian solvable group. It turns out that any polycyclic group is finitely presented.
• Finitely generated abelian implies finitely presented (via polycyclic)
• Finitely generated nilpotent implies finitely presented (via polycyclic)

Proof

Example of the wreath product of group of integers with group of integers

Further information: Wreath product of group of integers with group of integers