# Projective special linear group of degree two

## Contents

## Definition

### For a field or commutative unital ring

The **projective special linear group of degree two** over a field , or more generally over a commutative unital ring , is defined as the quotient of the special linear group of degree two over the same field or commutative unital ring by the subgroup of scalar matrices in that group. The group is denoted by or .

### For a prime power

Suppose is a prime power. The projective special linear group is defined as the projective special linear group of degree two over the field (unique up to isomorphism) with elements.

## Particular cases

### For prime powers

Note that for , . Also, for a power of 2 (so ), and .

## Arithmetic functions

Below we give the arithmetic functions for , where is a power of a prime .

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

order | Becomes for odd , when |
See order formulas for linear groups of degree two. Also, see element structure of projective special linear group of degree two |

number of conjugacy classes | for odd , when | See element structure of projective special linear group of degree two |

## Group properties

The property listings below are for , a prime power.

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

abelian group | No (never) | |

nilpotent group | No (never) | |

solvable group | No (never) | |

simple group, simple non-abelian group | Yes (almost always) | Exceptions: (we get symmetric group:S3) and (we get alternating group:A4). See projective special linear group is simple. |

minimal simple group | Sometimes | See classification of finite minimal simple groups. Minimal simple in precisely these cases: , prime; , an odd prime; is a prime greater than 3 such that divides , and (which is not necessary to add, since so it gets double-counted). In particular, the following values of give simple non-abelian groups that are not minimal simple: (gives alternating group:A6), . |

## Elements

### Over a finite field

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

Below is a summary:

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

conjugacy class sizes | Case congruent to 1 mod 4 (e.g., ): 1 (1 time), (2 times), (1 time), ( times), ( times) Case congruent to 3 mod 4 (e.g., : 1 (1 time), (1 time), (2 times), ( times), ( times Case even (e.g., ): 1 (1 time), ( times), (1 time), ( times) |

number of conjugacy classes | Case odd : Case even: equals number of irreducible representations, see also linear representation theory of projective special linear group of degree two over a finite field |

number of -regular conjugacy classes (where is the characteristic of the field) | equals the number of irreducible representations in that characteristic, see also modular representation theory of projective special linear group of degree two over a finite field in its defining characteristic |

order | General formula: Case odd: Case even: |

exponent | Case odd: Case even: |

## Linear representation theory

### Over a finite field

`Further information: linear representation theory of projective special linear group of degree two over a finite field`

Below is a summary:

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

degrees of irreducible representations over a splitting field (such as or ) | Case congruent to 1 mod 4 (e.g., ): 1 (1 time), (2 times), ( times), (1 time), ( times) Case congruent to 3 mod 4 (e.g., ): 1 (1 time), (2 times), ( times), (1 time), ( times) Case even (e.g., ): 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 projective special linear group of degree two over a finite field#Conjugacy class structure |

quasirandom degree (minimum possible degree of nontrivial irreducible representation) | Case congruent to 1 mod 4 (e.g., ): Case congruent to 3 mod 4 (e.g., ): Case even: |

maximum degree of irreducible representation over a splitting field | |

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

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