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

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

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