# Special linear group of degree two

## Contents

## Definition

### For a field or commutative unital ring

The **special linear group of degree two** over a field , or more generally over a commutative unital ring , is defined as the group of matrices with determinant under matrix multiplication, and entries over . The group is denoted by or .

When is a prime power, is the special linear group of degree two over the field (unique up to isomorphism) with elements.

The underlying set of the group is:

.

The group operation is given by:

.

The identity element is:

.

The inverse map is given by:

### For a prime power

Let be a prime power. The special linear group is defined as , where is the (unique up to isomorphism) field of size .

Note that for a finite field, we have the following: special unitary group of degree two equals special linear group of degree two over a finite field. In other words, is isomorphic to .

## Arithmetic functions

### Over a finite field

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 | Similar groups | Explanation |
---|---|---|---|

order | The projective general linear group of degree two has the same order, but is not isomorphic to it unless is a power of 2. | Kernel of determinant map from , a group of size surjecting to , a group of size . The order is thus . See order formulas for linear groups of degree two for more information. | |

exponent | if , if | There are elements of order , orders of all elements divide one of these. | |

number of conjugacy classes | if , if | For , semisimple conjugacy classes (that do not split from and four conjugacy classes that merge into two in . |

## Group properties

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

Abelian group | Yes if , no otherwise | |

Nilpotent group | Yes if , no otherwise | special linear group is perfect for , the case of can be checked. |

Solvable group | Yes if , no otherwise. | special linear group is perfect for , the case of can be checked. |

Supersolvable group | Yes if , no otherwise | special linear group is perfect for , the case of can be checked. |

Quasisimple group | Yes if | special linear group is quasisimple for . |

## Elements

### Over a finite field

`Further information: Element structure of special linear group of degree two over a finite field`

As before, is the field size and is the characteristic of the field, so is a prime number and is a power of .

## Linear representation theory

### Over a finite field

`Further information: Linear representation theory of special linear group of degree two over a finite field, modular representation theory of special linear group of degree two over a finite field in its defining characteristic`

Here is a summary of the linear representation theory in characteristic zero (and all characteristics coprime to the order):

Item | Value |
---|---|

degrees of irreducible representations over a splitting field (such as or ) | Case odd: 1 (1 time), (2 times), (2 times), ( times), (1 time), ( times) Case even: 1 (1 time), ( times), (1 time), ( times) |

number of irreducible representations | Case odd: Case even: See number of irreducible representations equals number of conjugacy classes, element structure of special linear group of degree two over a finite field#Conjugacy class structure |

quasirandom degree (minimum degree of nontrivial irreducible representation) | Case odd: Case even: |

maximum degree of irreducible representation over a splitting field | if if |

lcm of degrees of irreducible representations over a splitting field | Case : We get 6 Case odd, : Case even: |

sum of squares of degrees of irreducible representations over a splitting field | , equal to the group order. See sum of squares of degrees of irreducible representations equals group order |

Here is a summary of the modular representation theory in characteristic , where is the characteristic of the field over which we are taking the special linear group (so is a power of ):