# Element structure of special linear group of degree three over a finite field

This article gives specific information, namely, element structure, about a family of groups, namely: special linear group of degree three.

View element structure of group families | View other specific information about special linear group of degree three

This article describes the element structure of the special linear group of degree three over a finite field.

We take as the number of elements in the field and as the underlying prime number, so is a power of .

## Summary

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

number of conjugacy classes | Case is not 1 mod 3 (e.g., ): Case (e.g., ): |

order | |

exponent | ? |

## Conjugacy class structure

### Number of conjugacy classes

As we know in general, number of conjugacy classes in special linear group of fixed degree over a finite field is PORC function of field size, the degree of this PORC function is one less than the degree of matrices, and we make cases based on the congruence classes modulo the degree of matrices. Thus, we expect that the number of conjugacy classes is a PORC function of the field size of degree 3 - 1 = 2, and we need to make cases based on the congruence class of the field size modulo 3. Moreover, the general theory also tells us that the polynomial function of depends *only* on the value of , which in turn can be determined by the congruence class of mod (with here).

Value of | Corresponding congruence classes of mod 3 | Number of conjugacy classes (polynomial of degree 3 - 1 = 2 in ) | Additional comments |
---|---|---|---|

1 | 0 or -1 mod 3: 0 mod 3 (e.g. ) -1 mod 3 (e.g., ) |
In this case, we have an isomorphism between linear groups when degree power map is bijective, so | |

3 | 1 mod 3 (e.g., ) |

### General strategy and summary

`Further information: Element structure of special linear group over a finite field, conjugacy class of elements with semisimple generalized Jordan block does not split in special linear group over a finite field, splitting criterion for conjugacy classes in special linear group of prime degree over a finite field`

Before making the entire table, we recall the general strategy: first, imitate the procedure of element structure of general linear group of degree three over a finite field to determine that -conjugacy classes in . Then, use the splitting criterion for conjugacy classes in the special linear group to determine which of these conjugacy classes split, and how much.

In this case, the fact that conjugacy class of elements with semisimple generalized Jordan block does not split in special linear group over a finite field tells us that the *only* conjugacy class that *might* split is the conjugacy class with a Jordan block of size three. Further, the splitting criterion for conjugacy classes in special linear group of prime degree over a finite field, applied to the case of degree three, tells us that:

- When the field size is 1 mod 3, there are three such -conjugacy classes (corresponding to the cube roots of unity) and each splits into three -conjugacy classes. We thus get a total of 9 -conjugacy classes of this type.
- When the field size is not 1 mod 3, there is only one such -conjugacy class and it does not split over .

### Summary for field size 1 mod 3

The key feature for fields of size is that such fields have three distinct cube roots of unity.

Nature of conjugacy class | Eigenvalues | Characteristic polynomial | Minimal polynomial | Size of conjugacy class | Number of such conjugacy classes | Total number of elements | Semisimple? | Diagonalizable over ? | Splits in relative to ? |
---|---|---|---|---|---|---|---|---|---|

Diagonalizable over with equal diagonal entries, hence a scalar | where | for the eigenvalue | 1 | 3 | 3 | Yes | Yes | No | |

Diagonalizable over with one eigenvalue having multiplicity two, other eigenvalue having multiplicity one | where but is not a cube root of unity | Yes | Yes | No | |||||

Diagonalizable over with all distinct diagonal entries | all distinct, with | same as characteristic polynomial | Yes | Yes | No | ||||

Diagonalizable over , not over | Distinct Galois conjugate triple of elements in of norm 1. If one of the elements is , the other two are and . We must have . | irreducible degree three polynomial over | same as characteristic polynomial | Yes | No | No | |||

One eigenvalue is in , the other two are in | one element of , pair of Galois conjugates over in , whose norm is reciprocal of that element | product of linear polynomial and irreducible degree two polynomial over | same as characteristic polynomial | Yes | No | No | |||

Has Jordan blocks of sizes 2 and 1 with distinct eigenvalues over | with , not a cube root of unity | same as characteristic polynomial | No | No | No | ||||

Has Jordan blocks of sizes 2 and 1 with equal eigenvalues over | with | 3 | No | No | No | ||||

Has Jordan block of size 3 | with | same as characteristic polynomial | 9 | No | No | Yes | |||

Total (--) | -- | -- | -- | -- | -- | -- | -- |

### Summary for field size not 1 mod 3

In this case, the only cube root of 1 is 1. In particular, the field does not have primitive cube roots of unity.

Nature of conjugacy class | Eigenvalues | Characteristic polynomial | Minimal polynomial | Size of conjugacy class | Number of such conjugacy classes | Total number of elements | Semisimple? | Diagonalizable over ? | Splits in relative to ? |
---|---|---|---|---|---|---|---|---|---|

Diagonalizable over with equal diagonal entries, hence a scalar | 1 | 1 | 1 | Yes | Yes | No | |||

Diagonalizable over with one eigenvalue having multiplicity two, other eigenvalue having multiplicity one | where is not 1 | Yes | Yes | No | |||||

Diagonalizable over with all distinct diagonal entries | all distinct, with | same as characteristic polynomial | Yes | Yes | No | ||||

Diagonalizable over , not over | Distinct Galois conjugate triple of elements in of norm 1. If one of the elements is , the other two are and . We must have . | irreducible degree three polynomial over | same as characteristic polynomial | Yes | No | No | |||

One eigenvalue is in , the other two are in | one element of , pair of Galois conjugates over in , whose norm is reciprocal of that element | product of linear polynomial and irreducible degree two polynomial over | same as characteristic polynomial | Yes | No | No | |||

Has Jordan blocks of sizes 2 and 1 with distinct eigenvalues over | with , not 1 | same as characteristic polynomial | No | No | No | ||||

Has Jordan blocks of sizes 2 and 1 with equal eigenvalues over | 1 | No | No | No | |||||

Has Jordan block of size 3 | same as characteristic polynomial | 1 | No | No | Yes | ||||

Total (--) | -- | -- | -- | -- | -- | -- | -- |