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

From Groupprops
Jump to: navigation, search
(Definition)
(GAP implementation)
Line 13: Line 13:
 
==GAP implementation==
 
==GAP implementation==
  
The group can be defined as:
+
The group can be defined using GAP's [[GAP:SpecialLinearGroup|SpecialLinearGroup]] as:
  
 
<pre>SL(2,Integers)</pre>
 
<pre>SL(2,Integers)</pre>

Revision as of 00:48, 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
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.

GAP implementation

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

SL(2,Integers)