# Difference between revisions of "General linear group of degree two"

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+ | [[importance rank::2| ]] | ||

==Definition== | ==Definition== | ||

− | + | ===For a unital ring=== | |

− | For a prime power <math>q</math>, <math>GL(2,q)</math> or <math>GL_2(q)</math> denotes the general linear group of degree two over the field (unique up to isomorphism) with <math>q</math> elements. | + | The '''general linear group of degree two''' over a unital ring <math>R</math> is defined as the group, under matrix multiplication, of invertible <math>2 \times 2</math> matrices with entries in <math>R</math>. It is denoted <math>GL(2,R)</math>. |

+ | |||

+ | ===For a commutative unital ring=== | ||

+ | |||

+ | When <math>R</math> is a commutative unital ring, a <math>2 \times 2</math> matrix over <math>R</math> being invertible is equivalent to its determinant being an invertible element of <math>R</math>, so the general linear group <math>GL(2,R)</math> is defined as the following group of matrices under matrix multiplication: | ||

+ | |||

+ | <math>GL(2,R) := \left \{ \begin{pmatrix} a & b \\ c & d \\\end{pmatrix} \mid a,b,c,d \in R, ad - bc \mbox{ is an invertible element of } R \right \}</math> | ||

+ | |||

+ | ===For a field=== | ||

+ | |||

+ | For a [[field]] <math>K</math>, an element is invertible iff it is nonzero, so the general linear group <math>GL(2,K)</math> is defined as the following group of matrices under matrix multiplication: | ||

+ | |||

+ | <math>GL(2,K) := \left \{ \begin{pmatrix} a & b \\ c & d \\\end{pmatrix} \mid a,b,c,d \in K, ad - bc \ne 0 \right \}</math> | ||

+ | |||

+ | ===For a prime power=== | ||

+ | |||

+ | For a [[prime power]] <math>q</math>, <math>GL(2,q)</math> or <math>GL_2(q)</math> denotes the general linear group of degree two over the [[finite field]] (unique up to isomorphism) with <math>q</math> elements. This is a field of characteristic <math>p</math>, where <math>p</math> is the [[prime number]] whose power is <math>q</math>. | ||

==Particular cases== | ==Particular cases== | ||

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! Function !! Value !! Explanation | ! Function !! Value !! Explanation | ||

|- | |- | ||

− | | [[order of a group|order]] || <math>\! (q^2 - 1)(q^2 - q) = q^4 - q^3 - q^2 + q = q(q + 1)(q-1)^2</math> || <math>q^2 - 1</math> options for first row, <math>q^2 - q</math> options for second row.<br>See [[order formulas for linear groups]] | + | | [[order of a group|order]] || <math>\! (q^2 - 1)(q^2 - q) = q^4 - q^3 - q^2 + q = q(q + 1)(q-1)^2</math> || <math>q^2 - 1</math> options for first row, <math>q^2 - q</math> options for second row.<br>See [[order formulas for linear groups of degree two]] |

|- | |- | ||

| [[exponent of a group|exponent]] || <math>\! p(q^2 - 1) = p(q-1)(q+1)</math> || There is an element of order <math>q^2 - 1</math> and an element of order <math>p(q - 1)</math>. All elements have order dividing <math>p(q - 1)</math> or <math>q^2 - 1</math>. | | [[exponent of a group|exponent]] || <math>\! p(q^2 - 1) = p(q-1)(q+1)</math> || There is an element of order <math>q^2 - 1</math> and an element of order <math>p(q - 1)</math>. All elements have order dividing <math>p(q - 1)</math> or <math>q^2 - 1</math>. | ||

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==Elements== | ==Elements== | ||

− | { | + | ===Information based on ring type=== |

+ | |||

+ | {| class="sortable" border="1" | ||

+ | ! Ring type !! Element structure page | ||

+ | |- | ||

+ | | [[field]] || [[element structure of general linear group of degree two over a field]] | ||

+ | |- | ||

+ | | [[finite field]] || [[element structure of general linear group of degree two over a finite field]] | ||

+ | |- | ||

+ | | [[finite discrete valuation ring]] || [[element structure of general linear group of degree two over a finite discrete valuation ring]] | ||

+ | |- | ||

+ | | [[division ring]] || [[element structure of general linear group of degree two over a division ring]] | ||

+ | |} | ||

+ | ===Conjugacy class structure (case of a field)=== | ||

− | {{#lst:element structure of general linear group of degree two over a | + | {{#lst:element structure of general linear group of degree two over a field|conjugacy class structure}} |

==Subgroup-defining functions== | ==Subgroup-defining functions== |

## Latest revision as of 21:13, 18 September 2012

## Contents

## Definition

### For a unital ring

The **general linear group of degree two** over a unital ring is defined as the group, under matrix multiplication, of invertible matrices with entries in . It is denoted .

### For a commutative unital ring

When is a commutative unital ring, a matrix over being invertible is equivalent to its determinant being an invertible element of , so the general linear group is defined as the following group of matrices under matrix multiplication:

### For a field

For a field , an element is invertible iff it is nonzero, so the general linear group is defined as the following group of matrices under matrix multiplication:

### For a prime power

For a prime power , or denotes the general linear group of degree two over the finite field (unique up to isomorphism) with elements. This is a field of characteristic , where is the prime number whose power is .

## Particular cases

### Finite fields

Common name for general linear group of degree two | Field | Size of field | Order of group |
---|---|---|---|

symmetric group:S3 | field:F2 | 2 | 6 |

general linear group:GL(2,3) | field:F3 | 3 | 48 |

direct product of A5 and Z3 | field:F4 | 4 | 180 |

general linear group:GL(2,5) | field:F5 | 5 | 480 |

### Infinite rings and fields

Name of ring/field | Common name for general linear group of degree two |
---|---|

Ring of integers | general linear group:GL(2,Z) |

Field of rational numbers | general linear group:GL(2,Q) |

Field of real numbers | general linear group:GL(2,R) |

Field of complex numbers | general linear group:GL(2,C) |

## Arithmetic functions

Here, denotes the order of the finite field and the group we work with is . is the characteristic of the field, i.e., it is the prime whose power is.

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

order | options for first row, options for second row. See order formulas for linear groups of degree two | |

exponent | There is an element of order and an element of order . All elements have order dividing or . | |

number of conjugacy classes | There are conjugacy classes of semisimple matrices and conjugacy classes of matrices with repeated eigenvalues. |

## Group properties

Property | Satisfied? | Explanation |
---|---|---|

abelian group | No | The matrices and don't commute. |

nilpotent group | No | is simple for , and we can check the cases separately. |

solvable group | Yes if , no otherwise. | is simple for . |

supersolvable group | Yes if , no otherwise. | is simple for , and we can check the cases separately. |

## Elements

### Information based on ring type

### Conjugacy class structure (case of a field)

Nature of conjugacy class | Eigenvalues | Characteristic polynomial | Minimal polynomial | What set can each conjugacy class be identified with? (rough measure of size of conjugacy class) | What can the set of conjugacy classes be identified with (rough measure of number of conjugacy classes) | What can the union of conjugacy classes be identified with? | Semisimple? | Diagonalizable over ? |
---|---|---|---|---|---|---|---|---|

Diagonalizable over with equal diagonal entries, hence a scalar | where | where | where | one-point set | Yes | Yes | ||

Diagonalizable over with distinct diagonal entries |
(interchangeable) distinct elements of | Same as characteristic polynomial | set of decompositions of a fixed two-dimensional vector space over as a direct sum of one-dimensional subspaces | the set | ? | Yes | Yes | |

Diagonalizable over a quadratic extension of but not over itself. Must necessarily have no repeated eigenvalues. | Pair of conjugate elements of some separable quadratic extension of | , irreducible | Same as characteristic polynomial | ? | ? | ? | Yes | No |

Not diagonal, has Jordan block of size two with eigenvalue in | (multiplicity two) where | where | Same as characteristic polynomial | ? | ? | ? | No | No |

Not diagonal, has Jordan block of size two with eigenvalue not in |
(multiplicity two) where is in a purely inseparable quadratic extension of , so . This case arises only when has characteristic two and is not a perfect field | , a non-square in , has characteristic two | Same as characteristic polynomial | ? | ? | ? | No | No |

## Subgroup-defining functions

Subgroup-defining function | Value | Explanation |
---|---|---|

Center | The subgroup of scalar matrices. Cyclic of order | Center of general linear group is group of scalar matrices over center. |

Commutator subgroup | Except the case of , it is the special linear group of degree two, which has index . | Commutator subgroup of general linear group is special linear group |

## Quotient-defining functions

Subgroup-defining function | Value | Explanation |
---|---|---|

Inner automorphism group | Projective general linear group of degree two | Quotient by the center, which is the group of scalar matrices. |

Abelianization | This is isomorphic to the multiplicative group of the field. | Quotient by the commutator subgroup, which is the special linear group, which is the kernel of the determinant map that surjects to the multiplicative group of the field. |