Noetherian not implies finitely presented

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 fact about::group in which every subgroup is finitely presented.

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.

Example of the Tarski group
For a sufficiently large prime number $$p$$, 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 $$p$$. Clearly, the group is Noetherian and in fact has subgroup rank $$2$$: the whole group needs two generators and every proper nontrivial subgroup needs $$1$$ generator.

However, Tarski monsters are not finitely presented.