Polycyclic group: Difference between revisions

Latest revision as of 01:48, 29 March 2013

Definition

A group is said to be polycyclic if it satisfies the following equivalent conditions:

It has a subnormal series (wherein each member is normal in its successor) of finite length, starting from the trivial group and ending at the whole group, such that all the successive quotients are cyclic.
It is a solvable group and all the factor groups between successive members of its derived series are finitely generated abelian groups.
It is both a Noetherian group (also called a slender group, i.e., every subgroup is finitely generated) and a solvable group.

Definition with symbols

A group $G$ is said to be polycyclic if there exists a series of subgroups:

$1 = H_{0} ◃ H_{1} ◃ H_{2} ◃ \dots ◃ H_{n} = G$

where each $H_{i + 1} / H_{i}$ is cyclic.

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 page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: Noetherian group and solvable group
View other group property conjunctions OR view all group properties

This is a variation of solvable group|Find other variations of solvable group |

This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
VIEW: Definitions built on this | Facts about this: (facts closely related to Polycyclic group, all facts related to Polycyclic group) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
View a complete list of semi-basic definitions on this wiki

History

Origin of the concept

Hirsch, in the years 1938-1954, obtained results on polycylic groups, but he used the term S-group for them.

Origin of the term

The term polycyclic group was first used by Hall in 1954. It is now the commonly accepted term.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Cyclic group				\|FULL LIST, MORE INFO
Metacyclic group				\|FULL LIST, MORE INFO
Supersolvable group				\|FULL LIST, MORE INFO
Finite solvable group				\|FULL LIST, MORE INFO
Finitely generated abelian group				\|FULL LIST, MORE INFO
Finitely generated nilpotent group				\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
solvable group			(any infinitely generated solvable group will do)	\|FULL LIST, MORE INFO
finitely generated group	has a generating set that is finite		(any finitely generated non-solvable group will do)	\|FULL LIST, MORE INFO
Noetherian group	every subgroup is finitely generated			\|FULL LIST, MORE INFO
finitely generated solvable group			finitely generated and solvable not implies polycyclic	\|FULL LIST, MORE INFO
finitely presented group	has a presentation that uses a finite number of generators and a finite number of relations	polycyclic implies finitely presented	(any finitely presented non-solvable group will do)	\|FULL LIST, MORE INFO
finitely presented solvable group	both finitely presented and solvable	(via separate implications for finitely presented and solvable)	finitely presented and solvable not implies polycyclic	\|FULL LIST, MORE INFO

Metaproperties

Subgroups

This group property is subgroup-closed, viz., any subgroup of a group satisfying the property also satisfies the property
View a complete list of subgroup-closed group properties

Any subgroup of a polycyclic group is polycyclic. The subnormal serise for the subgroup is obtained by intersecting with it the subnormal series of the whole group.

Quotients

This group property is quotient-closed, viz., any quotient of a group satisfying the property also has the property
View a complete list of quotient-closed group properties

Any quotient of a polycyclic group is polycyclic. The subnormal series for the quotient is obtained by taking the image of the subnormal series for the whole group, via the quotient map.

Direct products

This group property is finite direct product-closed, viz the direct product of a finite collection of groups each having the property, also has the property
View other finite direct product-closed group properties

A finite direct product of polycyclic groups is polycyclic.

@@ Line 1: / Line 1: @@
-{{group property}}
+==Definition==
+A group is said to be '''polycyclic''' if it satisfies the following equivalent conditions:
+# It has a [[defining ingredient::subnormal series]] (wherein each member is normal in its successor) of finite length, starting from the trivial group and ending at the whole group, such that all the successive quotients are [[defining ingredient::cyclic group|cyclic]].
+# It is a [[defining ingredient::solvable group]] and all the factor groups between successive members of its [[derived series]] are [[defining ingredient::finitely generated abelian group]]s.
+# It is both a [[defining ingredient::Noetherian group]] (also called a slender group, i.e., every subgroup is finitely generated) and a [[defining ingredient::solvable group]].
+===Definition with symbols===
+A group <math>G</math> is said to be '''polycyclic''' if there exists a series of subgroups:
+<math>1 = H_0 \triangleleft H_1 \triangleleft H_2 \triangleleft \ldots \triangleleft H_n = G</math>
-{{variationof|solvability}}
+where each <math>H_{i+1}/H_i</math> is [[cyclic group|cyclic]].
+{{group property}}
+{{group property conjunction|Noetherian group|solvable group}}
+{{variation of|solvable group}}
 {{semibasicdef}}
@@ Line 15: / Line 30: @@
 The term '''polycyclic group''' was first used by [[Hall]] in 1954. It is now the commonly accepted term.
-==Definition==
-A group is said to be '''polycyclic''' if it has a [[subnormal series]] (wherein each member is normal in its successor) of finite length, starting from the trivial group and ending at the whole group, such that all the successive quotients are [[cyclic group|cyclic]].
-===Definition with symbols===
-A group <math>G</math> is said to be '''polycyclic''' if there exists a series of subgroups:
-<math>1 = H_0 \triangleleft H_1 \triangleleft H_2 \triangleleft \ldots \triangleleft H_n = G</math>
-where each <math>H_{i+1}/H_i</math> is [[cyclic group|cyclic]].
 ==Relation with other properties==
@@ Line 52: / Line 57: @@
 ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
 |-
-| [[Stronger than::Solvable group]] || || || || {{intermediate notions short|solvable group|polycyclic group}}
+| [[Stronger than::solvable group]] || || || (any infinitely generated solvable group will do) || {{intermediate notions short|solvable group|polycyclic group}}
+|-
+| [[Stronger than::finitely generated group]] || has a [[generating set of a group|generating set]] that is finite || || (any finitely generated non-solvable group will do) || {{intermediate notions short|finitely generated group|polycyclic group}}
+|-
+| [[Stronger than::Noetherian group]] || every subgroup is finitely generated || || || {{intermediate notions short|slender group|polycyclic group}}
 |-
-| [[Stronger than::Finitely generated group]] || || || || {{intermediate notions short|finitely generated group|polycyclic group}}
+| [[Stronger than::finitely generated solvable group]] || || || [[finitely generated and solvable not implies polycyclic]] || {{intermediate notions short|finitely generated solvable group|polycyclic group}}
 |-
-| [[Stronger than::Slender group]] || || || || {{intermediate notions short|slender group|polycyclic group}}
+| [[Stronger than::finitely presented group]] || has a [[presentation of a group|presentation]] that uses a finite number of generators and a finite number of relations || [[polycyclic implies finitely presented]] || (any finitely presented non-solvable group will do) || {{intermediate notions short|finitely presented group|polycyclic group}}
 |-
-| [[Stronger than::Finitely generated solvable group]] || || || || {{intermediate notions short|finitely generated solvable group|polycyclic group}}
+| [[Stronger than::finitely presented solvable group]] || both finitely presented and solvable || (via separate implications for finitely presented and solvable) || [[finitely presented and solvable not implies polycyclic]] || {{intermediate notions short|finitely presented solvable group|polycyclic group}}
 |}