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

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The group also has the following equivalent descriptions:
 
The group also has the following equivalent descriptions:
  
* The [[inner automorphism group]] of [[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]].
 
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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).
 
==Arithmetic functions==
 
==Arithmetic functions==
  

Revision as of 22:14, 11 July 2011

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:

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)