# Noetherian 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., Noetherian group) neednotsatisfy the second group property (i.e., finitely presented group)

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## Contents

## Statement

A Noetherian group (also called *slender group*, and defined as a group in which every subgroup is finitely generated) need not be a finitely presented group, i.e., it need not possess any finite presentation. In particular, it need not be a Group in which every subgroup is finitely presented (?).

## Related facts

### Similar facts

- Finitely generated not implies finitely presented
- Finitely generated and solvable not implies finitely presented

### Opposite facts

- Polycyclic implies finitely presented: A Noetherian solvable group is a polycyclic group, and any polycyclic group is finitely presented. Since solvability is subgroup-closed, every subgroup of a polycyclic group is finitely presented. In particular, any finitely generated abelian group and any finitely generated nilpotent group is finitely presented.

## Proof

### Example of the Tarski group

`Further information: Tarski group, Tarski group is not finitely presented`

For a sufficiently large prime number , we can construct a Tarski group (also called *Tarski monster*): an infinite simple group in which the only proper nontrivial subgroups are cyclic of order . Clearly, the group is Noetherian and in fact has subgroup rank : the whole group needs two generators and every proper nontrivial subgroup needs generator.

However, Tarski monsters are not finitely presented.