Finitely generated and solvable not implies finitely presented

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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

Opposite facts

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