Difference between revisions of "Special linear group:SL(2,Z)"

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This is the degree two case of a [[member of family::special linear group over integers]] and hence of a [[member of family::special linear group]]. It is also a special case of a [[member of family::special linear group of degree two]].
 
This is the degree two case of a [[member of family::special linear group over integers]] and hence of a [[member of family::special linear group]]. It is also a special case of a [[member of family::special linear group of degree two]].
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The group also has the following equivalent descriptions:
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* The [[defining ingredient::amalgamated free product]] of [[defining ingredient::cyclic group:Z4]] and [[defining ingredient::cyclic group:Z6]] over amalgamated subgroup [[cyclic group:Z2]] (living as [[Z2 in Z4]] and [[Z2 in Z6]] respectively).
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===Definition by presentation===
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The group can be defined by any of the following presentations (here, <math>1</math> denotes the identity element):
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* {{fillin}}
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* From the amalgamated free product definition: <math>\langle x,y \mid x^4 = 1, x^2 = y^3 \rangle</math>
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===Structures===
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Thinking of <math>SL(2,\mathbb{Z})</math> as a group of matrices, we see that it is an example of an [[has structure of::arithmetic group]].
  
 
==Arithmetic functions==
 
==Arithmetic functions==
  
{| class="wikitable" border="1"
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{| class="sortable" border="1"
 
! Function !! Value !! Explanation
 
! Function !! Value !! Explanation
 
|-
 
|-
| [[order of a group|order]] || infinite (countable) ||
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| [[order of a group|order]] || infinite (countable) || As <math>SL(2,\mathbb{Z})</math>: The group is infinite because, for instance, it contains all matrices of the form <math>\begin{pmatrix} 1 &  a \\ 0 & 1 \\\end{pmatrix}</math> for <math>a \in \mathbb{Z}</math>. <br>As a set, the group is contained in the set of all <math>2 \times 2</math> matrices over <math>\mathbb{Z}</math>. This can be identified with <math>\mathbb{Z}^4</math>, which is countable since <math>\mathbb{Z}</math> is countable. Thus, <math>SL(2,\mathbb{Z})</math> is also countable.<br>As an amalgamated free product: any amalgamated free product relative to a subgroup that is proper in both groups is infinite.
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|-
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| [[exponent of a group|exponent]] || infinite  || As <math>SL(2,\mathbb{Z})</math>: The group contains the element <math>\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix}</math>, which has infinite [[order of an element|order]].
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|-
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| {{arithmetic function value|minimum size of generating set|2}} || As <math>SL(2,\mathbb{Z})</math>: Follows from [[elementary matrices of the first kind generate the special linear group over a Euclidean ring]], so <math>SL(2,\mathbb{Z})</math> is generated by all matrices of the form <math>\begin{pmatrix} 1 & a \\ 0 & 1 \\\end{pmatrix}</math> and <math>\begin{pmatrix} 1 & 0 \\ b & 1 \\\end{pmatrix}</math> with <math>a,b</math> varying over <math>\mathbb{Z}</math>. By the fact that the additive group of <math>\mathbb{Z}</math> is cyclic, all these matrices are generated by <math>\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix}</math> and <math>\begin{pmatrix} 1 & 0 \\ 1 & 1 \\\end{pmatrix}</math>.<br>As an amalgamated free product of two cyclic groups: follows that it is 2, just from the definition (note that the generators here are different from those used in the justification in matrix terms).
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|-
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| [[subgroup rank]] || infinite (countable) || <math>SL(2,\mathbb{Z})</math> has a subgroup that is isomorphic to [[free group:F2]] (see [[Sanov subgroup in SL(2,Z) is free of rank two]]). This in turn has free subgroups of countable rank.
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|}
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==Group properties==
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{| class="sortable" border="1"
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! Property !! Satisfied? !! Explanation !! Corollary properties satisfied/dissatisfied
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|-
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| [[satisfies property::2-generated group]] || Yes || See explanation for minimum size of generating set above || satisfies: [[satisfies property::finitely generated group]], [[satisfies property::countable group]]
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|-
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| [[dissatisfies property::Noetherian group]] || No || See explanation for subgroup rank above ||
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|-
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| [[satisfies property::finitely presented group]] || Yes || Any of the definitions (<math>SL</math>, B_amalgamated free product) gives a finite presentation ||
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|-
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| [[dissatisfies property::solvable group]] || No || contains subgroup isomorphic to [[free group:F2]] -- see [[Sanov subgroup in SL(2,Z) is free of rank two]] || dissatisfies: [[dissatisfies property::nilpotent group]], [[dissatisfies property::abelian group]]
 
|-
 
|-
| [[exponent of a group|exponent]] || infinite (countable) ||
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| [[satisfies property::group satisfying no nontrivial identity]] || Yes ||contains subgroup isomorphic to [[free group:F2]] -- see [[Sanov subgroup in SL(2,Z) is free of rank two]] ||
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|-
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| [[satisfies property::SQ-universal group]] || Yes || contains subgroup isomorphic to [[free group:F2]] -- see [[Sanov subgroup in SL(2,Z) is free of rank two]] ||
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|-
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| [[satisfies property::residually finite group]] || Yes || The kernels of the homomorphisms <math>SL(2,\mathbb{Z}) \to SL(2,\mathbb{Z}/n\mathbb{Z})</math> for natural numbers <math>n</math> are normal subgroups of finite index and their intersection is trivial. || satisfies: [[satisfies property::finitely generated residually finite group]]
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|-
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| [[satisfies property::Hopfian group]] || Yes || Follows from [[finitely generated and residually finite implies Hopfian]] || satisfies: [[satisfies property::finitely generated Hopfian group]]
 
|}
 
|}
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==Elements==
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{{further|[[element structure of special linear group:SL(2,Z)]]}}
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{{#lst:element structure of special linear group:SL(2,Z)|summary}}
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==Facts==
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* [[Sanov subgroup in SL(2,Z) is free of rank two]]
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* The homomorphism <math>SL(2,\mathbb{Z}) \to SL(2,\mathbb{Z}/n\mathbb{Z})</math> is surjective for any natural number <math>n</math>.
  
 
==GAP implementation==
 
==GAP implementation==
  
The group can be defined using GAP's [[GAP:SpecialLinearGroup|SpecialLinearGroup]] as:
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{| class="sortable" border="1"
 
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! Description !! Functions used
<pre>SL(2,Integers)</pre>
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|-
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| <tt>SL(2,Integers)</tt> || [[GAP:SL|SL]], [[GAP:Integers|Integers]]
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|}

Latest revision as of 20:47, 18 September 2012

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
View a complete list of particular groups (this is a very huge list!)[SHOW MORE]

Definition

The group SL(2,\mathbb{Z}) is defined as the group, under matrix multiplication, of 2 \times 2 matrices over \mathbb{Z}, the ring of integers, having determinant 1.

In other words, it is the group with underlying set:

\left \{ \begin{pmatrix} a & b \\ c & d \\\end{pmatrix} \mid a,b,c,d \in \mathbb{Z}, ad - bc = 1 \right \}

This is the degree two case of a special linear group over integers and hence of a special linear group. It is also a special case of a special linear group of degree two.

The group also has the following equivalent descriptions:

Definition by presentation

The group can be defined by any of the following presentations (here, 1 denotes the identity element):

  • PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
  • From the amalgamated free product definition: \langle x,y \mid x^4 = 1, x^2 = y^3 \rangle

Structures

Thinking of SL(2,\mathbb{Z}) as a group of matrices, we see that it is an example of an arithmetic group.

Arithmetic functions

Function Value Explanation
order infinite (countable) As SL(2,\mathbb{Z}): The group is infinite because, for instance, it contains all matrices of the form \begin{pmatrix} 1 &  a \\ 0 & 1 \\\end{pmatrix} for a \in \mathbb{Z}.
As a set, the group is contained in the set of all 2 \times 2 matrices over \mathbb{Z}. This can be identified with \mathbb{Z}^4, which is countable since \mathbb{Z} is countable. Thus, SL(2,\mathbb{Z}) is also countable.
As an amalgamated free product: any amalgamated free product relative to a subgroup that is proper in both groups is infinite.
exponent infinite As SL(2,\mathbb{Z}): The group contains the element \begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix}, which has infinite order.
minimum size of generating set 2 As SL(2,\mathbb{Z}): Follows from elementary matrices of the first kind generate the special linear group over a Euclidean ring, so SL(2,\mathbb{Z}) is generated by all matrices of the form \begin{pmatrix} 1 & a \\ 0 & 1 \\\end{pmatrix} and \begin{pmatrix} 1 & 0 \\ b & 1 \\\end{pmatrix} with a,b varying over \mathbb{Z}. By the fact that the additive group of \mathbb{Z} is cyclic, all these matrices are generated by \begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix} and \begin{pmatrix} 1 & 0 \\ 1 & 1 \\\end{pmatrix}.
As an amalgamated free product of two cyclic groups: follows that it is 2, just from the definition (note that the generators here are different from those used in the justification in matrix terms).
subgroup rank infinite (countable) SL(2,\mathbb{Z}) has a subgroup that is isomorphic to free group:F2 (see Sanov subgroup in SL(2,Z) is free of rank two). This in turn has free subgroups of countable rank.

Group properties

Property Satisfied? Explanation Corollary properties satisfied/dissatisfied
2-generated group Yes See explanation for minimum size of generating set above satisfies: finitely generated group, countable group
Noetherian group No See explanation for subgroup rank above
finitely presented group Yes Any of the definitions (SL, B_amalgamated free product) gives a finite presentation
solvable group No contains subgroup isomorphic to free group:F2 -- see Sanov subgroup in SL(2,Z) is free of rank two dissatisfies: nilpotent group, abelian group
group satisfying no nontrivial identity Yes contains subgroup isomorphic to free group:F2 -- see Sanov subgroup in SL(2,Z) is free of rank two
SQ-universal group Yes contains subgroup isomorphic to free group:F2 -- see Sanov subgroup in SL(2,Z) is free of rank two
residually finite group Yes The kernels of the homomorphisms SL(2,\mathbb{Z}) \to SL(2,\mathbb{Z}/n\mathbb{Z}) for natural numbers n are normal subgroups of finite index and their intersection is trivial. satisfies: finitely generated residually finite group
Hopfian group Yes Follows from finitely generated and residually finite implies Hopfian satisfies: finitely generated Hopfian group

Elements

Further information: element structure of special linear group:SL(2,Z)


Facts

GAP implementation

Description Functions used
SL(2,Integers) SL, Integers