Vipul: /* Proof */

2010-05-27T18:00:20Z

Proof

← Older revision		Revision as of 18:00, 27 May 2010
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	==Proof==		==Proof==

	===Example of the Tarski ~~monster~~===		===Example of the Tarski group===

	{{further\|[[Particular example::Tarski ~~monster~~]], [[Tarski ~~monster~~ is not finitely presented]]}}		{{further\|[[Particular example::Tarski group]], [[Tarski group is not finitely presented]]}}

	For a sufficiently large [[prime number]] <math>p</math>, we can construct a [[Tarski ~~monster~~]]: an infinite simple group in which the only proper nontrivial subgroups are cyclic of order <math>p</math>. Clearly, the group is Noetherian and in fact has [[subgroup rank of a group\|subgroup rank]] <math>2</math>: the whole group needs two generators and every proper nontrivial subgroup needs <math>1</math> generator.		For a sufficiently large [[prime number]] <math>p</math>, 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 <math>p</math>. Clearly, the group is Noetherian and in fact has [[subgroup rank of a group\|subgroup rank]] <math>2</math>: the whole group needs two generators and every proper nontrivial subgroup needs <math>1</math> generator.

	However, Tarski monsters are not finitely presented.		However, Tarski monsters are not finitely presented.

Vipul: Created page with '{{group property non-implication| stronger = Noetherian group| weaker = finitely presented group}} ==Statement== A Noetherian group (also called ''slender group'', and defi...'

2010-05-27T17:56:23Z

Created page with '{{group property non-implication| stronger = Noetherian group| weaker = finitely presented group}} ==Statement== A Noetherian group (also called ''slender group'', and defi...'

New page

{{group property non-implication|
stronger = Noetherian group|
weaker = finitely presented group}}

==Statement==

A [[Noetherian group]] (also called ''slender group'', and defined as a group in which every subgroup is [[fact about::finitely generated group|finitely generated]]) need not be a [[finitely presented group]], i.e., it need not possess any finite [[presentation of a group|presentation]]. In particular, it need not be a [[fact about::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 monster===

{{further|[[Particular example::Tarski monster]], [[Tarski monster is not finitely presented]]}}

For a sufficiently large [[prime number]] <math>p</math>, we can construct a [[Tarski monster]]: an infinite simple group in which the only proper nontrivial subgroups are cyclic of order <math>p</math>. Clearly, the group is Noetherian and in fact has [[subgroup rank of a group|subgroup rank]] <math>2</math>: the whole group needs two generators and every proper nontrivial subgroup needs <math>1</math> generator.

However, Tarski monsters are not finitely presented.

Noetherian not implies finitely presented - Revision history

Vipul: /* Proof */

Vipul: Created page with '{{group property non-implication| stronger = Noetherian group| weaker = finitely presented group}} ==Statement== A Noetherian group (also called ''slender group'', and defi...'