Noetherian group: Difference between revisions

Revision as of 17:02, 25 May 2010

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

This is a variation of finiteness (groups)|Find other variations of finiteness (groups) |

Definition

Symbol-free definition

A group is said to be slender or Noetherian or to satisfy the maximum condition on subgroups if it satisfies the following equivalent conditions:

Every subgroup is finitely generated
Any ascending chain of subgroups stabilizes after a finite length
Any nonempty collection of subgroups has a maximal element: a member of that collection that is not contained in any other member of the collection.

Formalisms

In terms of the hereditarily operator

This property is obtained by applying the hereditarily operator to the property: finitely generated group
View other properties obtained by applying the hereditarily operator

Relation with other properties

Stronger properties

Property	Meaning	Intermediate notions
Finite group		slender group\|finite group}
Finitely generated abelian group	finitely generated and abelian	\|FULL LIST, MORE INFO
Finitely generated nilpotent group	finitely generated and nilpotent	\|FULL LIST, MORE INFO
Supersolvable group	has a normal series where all successive quotients are cyclic	\|FULL LIST, MORE INFO
Polycyclic group	slender and solvable	\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Finitely generated group				\|FULL LIST, MORE INFO
Maximal-covering group				\|FULL LIST, MORE INFO
Group satisfying ascending chain condition on subnormal subgroups				\|FULL LIST, MORE INFO
Group satisfying subnormal join property				\|FULL LIST, MORE INFO
Group satisfying ascending chain condition on normal subgroups				\|FULL LIST, MORE INFO
Hopfian group	every surjective endomorphism is an automorphism	Slender implies Hopfian	Hopfian not implies slender	\|FULL LIST, MORE INFO
Finitely generated Hopfian group	finitely generated and Hopfian	Slender implies Hopfian		\|FULL LIST, MORE INFO

@@ Line 21: / Line 21: @@
 ===Stronger properties===
-* [[Weaker than::Finite group]]
+{| class="sortable" border="1"
-* [[Weaker than::Finitely generated abelian group]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Weaker than::Finitely generated nilpotent group]]
+|-
+| [[Weaker than::Finite group]] || || || || {{intermediate notions short|slender group|finite group}
+|-
+| [[Weaker than::Finitely generated abelian group]] || [[finitely generated group|finitely generated]] and [[abelian group|abelian]] || || || {{intermediate notions short|slender group|finitely generated abelian group}}
+|-
+| [[Weaker than::Finitely generated nilpotent group]] || [[finitely generated group|finitely generated]] and [[nilpotent group|nilpotent]] || || || {{intermediate notions short|slender group|finitely generated nilpotent group}}
+|-
+| [[Weaker than::Supersolvable group]] || has a [[normal series]] where all successive quotients are cyclic || || || {{intermediate notions short|slender group|supersolvable group}}
+|-
+| [[Weaker than::Polycyclic group]] || slender and [[solvable group|solvable]] || || || {{intermediate notions short|slender group|polycyclic group}}
+|}
 ===Weaker properties===
-* [[Stronger than::Finitely generated group]]
+{| class="sortable" border="1"
-* [[Stronger than::Maximal-covering group]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Stronger than::Group satisfying ascending chain condition on subnormal subgroups]]
+|-
-* [[Stronger than::Group satisfying subnormal join property]]
+| [[Stronger than::Finitely generated group]] || || || || {{intermediate notions short|finitely generated group|slender group}}
-* [[Stronger than::Group satisfying ascending chain condition on normal subgroups]]
+|-
-* [[Stronger than::Hopfian group]]: {{proofofstrictimplicationat|[[Slender implies Hopfian]]|[[Hopfian not implies slender]]}}
+| [[Stronger than::Maximal-covering group]] || || || || {{intermediate notions short|maximal-covering group|slender group}}
-* [[Stronger than::Direct product of finitely many indecomposable groups]]
+|-
+| [[Stronger than::Group satisfying ascending chain condition on subnormal subgroups]] || || || || {{intermediate notions short|group satisfying ascending chain condition on subnormal subgroups|slender group}}
+|-
+| [[Stronger than::Group satisfying subnormal join property]] || || || || {{intermediate notions short|group satisfying subnormal join property|slender group}}
+|-
+| [[Stronger than::Group satisfying ascending chain condition on normal subgroups]] || || || || {{intermediate notions short|group satisfying ascending chain condition on subnormal subgroups|slender group}}
+|-
+| [[Stronger than::Hopfian group]] || every [[surjective endomorphism]] is an [[automorphism]] || [[Slender implies Hopfian]]||[[Hopfian not implies slender]] || {{intermediate notions short|Hopfian group|slender group}}
+|-
+| [[Stronger than::Finitely generated Hopfian group]] || finitely generated and Hopfian || [[Slender implies Hopfian]]|||| {{intermediate notions short|finitely generated Hopfian group|slender group}}
+|}