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) need not satisfy the second group property (i.e., finitely presented group)
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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 (?).
- Finitely generated not implies finitely presented
- Finitely generated and solvable not implies finitely presented
- 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 , 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.