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

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* The [[defining ingredient::inner automorphism group]] of [[defining ingredient::braid group:B3]], i.e., the quotient of <math>B_3</math> by its [[center]].
 
* The [[defining ingredient::inner automorphism group]] of [[defining ingredient::braid group:B3]], i.e., the quotient of <math>B_3</math> by its [[center]].
 
* 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).
 
* 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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===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 [[arithmetic group]].
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==Arithmetic functions==
 
==Arithmetic functions==
  

Revision as of 00:26, 11 August 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
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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:

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)
exponent infinite (countable)

GAP implementation

The group can be defined using GAP's SpecialLinearGroup as:

SL(2,Integers)