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

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(Created page with '{{particular group}} ==Definition== The group <math>SL(2,\mathbb{Z})</math> is defined as the group, under matrix multiplication, of <math>2 \times 2</math> matrices over <math…')
 
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In other words, it is the group with underlying set:
 
In other words, it is the group with underlying set:
  
<math>\{ \begin{pmatrix} a & b \\ c & d \\\end{pmatrix} \mid a,b,c,d \in \mathbb{Z}, ad - bc = 1 \}</math>
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<math>\left \{ \begin{pmatrix} a & b \\ c & d \\\end{pmatrix} \mid a,b,c,d \in \mathbb{Z}, ad - bc = 1 \right \}</math>
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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]].
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==GAP implementation==
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The group can be defined as:
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<pre>SL(2,Integers)</pre>

Revision as of 21:42, 24 August 2009

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.

GAP implementation

The group can be defined as:

SL(2,Integers)