Finitely presented not implies Noetherian

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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 presented group) need not satisfy the second group property (i.e., Noetherian group)
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A finitely presented group (i.e., a group that admits a finite presentation) need not be a Noetherian group (also called slender group: a group in which every subgroup is finitely generated).

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Example of the free group of rank two

Further information: Free group:F2

The free group of rank 2 is finitely presented (it has a presentation with two generators and no relations) but it is not Noetherian. For instance, in the free group \langle a, b \rangle with freely generating set \{ a, b \}, the normal closure of \langle a \rangle is a free group on infinitely many generators: \{ b^nab^{-n} \mid n \in \mathbb{Z} \}. Let's see why this is true.

Consider a free group on countably many generators x_n, n \in \mathbb{Z} and now consider an element y that acts on these generators by the rule x_n \mapsto x_{n + 1}. Consider the semidirect product of the free group on the x_ns with a cyclic group generated by y. Clearly, this group is generated by x_0,y and hence is a quotient of \langle a,b \rangle with a \mapsto x_0, b \mapsto y. We see thus that b^nab^{-n} \mapsto x_n and thus the normal closure of \langle a \rangle maps surjectively to the free group on the x_ns. Since the mapping is bijective on a generating set, we see that in fact the normal closure of \langle a \rangle is free on a countable generating set.

Example of the Baumslag-Solitar group

Further information: Baumslag-Solitar group:BS(1,2)

This group, given by the presentation:

\langle a,b \mid bab^{-1} = a^2 \rangle

is finitely presented, but the normal closure of \langle a \rangle in it is not finitely generated, since it is isomorphic to the additive group \mathbb{Z}[1/2] of 2-adic rationals.

Note that gives a solvable group example of the non-implication. See also finitely presented and solvable not implies polycyclic.