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

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− | | [[order of a group|order]] || infinite (countable) || | + | | [[order of a group|order]] || infinite (countable) || As <math>SL(2,\mathbb{Z})</math>: The group is infinite because, for instance, it contains all matrices of the form <math>\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix}</math> for <math>a \in \mathbb{Z}</math>. <br>As a set, the group is contained in the set of all <math>2 \times 2</math> matrices over <math>\mathbb{Z}</math>. This can be identified with <math>\mathbb{Z}^4</math>, which is countable since <math>\mathbb{Z}</math> is countable. Thus, <math>SL(2,\mathbb{Z})</math> is also countable.<br>As an amalgamated free product: any amalgamated free product relative to a subgroup that is proper in both groups is infinite. |

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− | | [[exponent of a group|exponent]] || infinite || | + | | [[exponent of a group|exponent]] || infinite || As <math>SL(2,\mathbb{Z})</math>: The group contains the element <math>\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix}</math>, which has infinite [[order of an element|order]].<br>As the inner automorphism group of the braid group: infinite, because the braid group generators have infinite order. |

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− | | {{arithmetic function value|minimum size of generating set|2}} || Follows from [[elementary matrices of the first kind generate the special linear group over a Euclidean ring]], so <math>SL(2,\mathbb{Z})</math> is generated by all matrices of the form <math>\begin{pmatrix} 1 & a \\ 0 & 1 \\\end{pmatrix}</math> and <math>1 & 0 \\ b & 1 \\\end{pmatrix}</math> with <math>a,b</math> varying over <math>\mathbb{Z}</math>. By the fact that the additive group of <math>\mathbb{Z}</math> is cyclic, all these matrices are generated by <math>\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix}</math> and <math>\begin{pmatrix} 1 & 0 \\ 1 & 1 \\\end{pmatrix}</math>. | + | | {{arithmetic function value|minimum size of generating set|2}} || As <math>SL(2,\mathbb{Z})</math>: Follows from [[elementary matrices of the first kind generate the special linear group over a Euclidean ring]], so <math>SL(2,\mathbb{Z})</math> is generated by all matrices of the form <math>\begin{pmatrix} 1 & a \\ 0 & 1 \\\end{pmatrix}</math> and <math>1 & 0 \\ b & 1 \\\end{pmatrix}</math> with <math>a,b</math> varying over <math>\mathbb{Z}</math>. By the fact that the additive group of <math>\mathbb{Z}</math> is cyclic, all these matrices are generated by <math>\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix}</math> and <math>\begin{pmatrix} 1 & 0 \\ 1 & 1 \\\end{pmatrix}</math>.<br>As the inner automorphism group of <math>B_3</math>: 2, because <math>B_3</math> itself is 2-generated. <br>As an amalgamated free product of two cyclic groups: follows that it is 2, just from the definition (note that the generators here are different from those used in the justification in matrix terms). |

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− | | [[subgroup rank]] || infinite (countable) | + | | [[subgroup rank]] || infinite (countable) || <math>SL(2,\mathbb{Z})</math> has a subgroup that is isomorphic to [[free group:F2]] (see [[Sanov subgroup in SL(2,Z) is free of rank two]]). This in turn has free subgroups of countable rank. |

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## Revision as of 01:03, 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 groupView a complete list of particular groups (this is a very huge list!)[SHOW MORE]

## Definition

The group is defined as the group, under matrix multiplication, of matrices over , the ring of integers, having determinant .

In other words, it is the group with underlying set:

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:

- The inner automorphism group of braid group:B3, i.e., the quotient of by its center.
- The amalgamated free product of cyclic group:Z4 and cyclic group:Z6 over amalgamated subgroup cyclic group:Z2 (living as Z2 in Z4 and Z2 in Z6 respectively).

### Structures

Thinking of as a group of matrices, we see that it is an example of an arithmetic group.

## Arithmetic functions

Function | Value | Explanation |
---|---|---|

order | infinite (countable) | As : The group is infinite because, for instance, it contains all matrices of the form for . As a set, the group is contained in the set of all matrices over . This can be identified with , which is countable since is countable. Thus, is also countable. As an amalgamated free product: any amalgamated free product relative to a subgroup that is proper in both groups is infinite. |

exponent | infinite | As : The group contains the element , which has infinite order. As the inner automorphism group of the braid group: infinite, because the braid group generators have infinite order. |

minimum size of generating set | 2 | As : Follows from elementary matrices of the first kind generate the special linear group over a Euclidean ring, so is generated by all matrices of the form and Failed to parse (syntax error): 1 & 0 \\ b & 1 \\\end{pmatrix}
with varying over . By the fact that the additive group of is cyclic, all these matrices are generated by and .As the inner automorphism group of : 2, because itself is 2-generated. As an amalgamated free product of two cyclic groups: follows that it is 2, just from the definition (note that the generators here are different from those used in the justification in matrix terms). |

subgroup rank | infinite (countable) | has a subgroup that is isomorphic to free group:F2 (see Sanov subgroup in SL(2,Z) is free of rank two). This in turn has free subgroups of countable rank. |

## GAP implementation

Description | Functions used |
---|---|

SL(2,Integers) |
SL, Integers |