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

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

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This article discusses the element structure of the general linear group of degree three over a finite field. The group is where is the order (size) of the field. We denote by the prime number that is the characteristic of the field.

## Particular cases

Group | Order of the group | Number of conjugacy classes | Element structure page | ||
---|---|---|---|---|---|

projective special linear group:PSL(3,2) | 2 | 2 | 168 | 6 | element structure of projective special linear group:PSL(3,2) |

general linear group:GL(3,3) | 3 | 3 | 11232 | 24 | element structure of general linear group:GL(3,3) |

general linear group:GL(3,5) | 5 | 5 | 1488000 | 120 | element structure of general linear group:GL(3,5) |

## Conjugacy class structure

There is a total of elements, and a total of conjugacy classes.

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 ? |
---|---|---|---|---|---|---|---|---|

Diagonalizable over with equal diagonal entries, hence a scalar | where | 1 | Yes | Yes | ||||

Diagonalizable over with one eigenvalue having multiplicity two, the other eigenvalue having multiplicity one | where , both in | Yes | Yes | |||||

Diagonalizable over with all distinct diagonal entries | , all distinct elements of | same as characteristic polynomial | Yes | Yes | ||||

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

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

Has Jordan blocks of sizes 2 and 1 with distinct eigenvalues over | with , | same as characteristic polynomial | No | No | ||||

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

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

Total | NA | NA | NA | NA | NA | NA |