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

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(GAP implementation)
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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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==Arithmetic functions==
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{| class="wikitable" border="1"
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! Function !! Value !! Explanation
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|-
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| [[order of a group|order]] || infinite (countable) ||
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|-
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| [[exponent of a group|exponent]] || infinite (countable) ||
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|}
  
 
==GAP implementation==
 
==GAP implementation==

Revision as of 01:18, 10 September 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. It is also a special case of a special linear group of degree two.

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)