# Conjugacy class of elements with semisimple generalized Jordan block does not split in special linear group over a finite field

## Statement

Suppose is a prime power and is a natural number. Then, there is a finite field of size , unique up to isomorphism. We denote the special linear group by . Similarly, we denote the general linear group by .

Suppose is an element of the special linear group and hence also of the general linear group . Suppose that, when is written in generalized Jordan block form, there is at least one generalized Jordan block that is semisimple, i.e., there are no repeated eigenvalues within the block. Then, the -conjugacy class of is precisely the same as the -conjugacy class of . In other words, the -conjugacy class of does not split in .

Note that the result applies in particular to the case where *itself* is semisimple (i.e., *all* its generalized Jordan blocks are semisimple) but it also applies in many cases where is not semisimple.

## Facts used

- Splitting criterion for conjugacy classes in the special linear group
- Norm map is surjective for finite fields

## Proof

**Given**: Natural number , prime power , element of whose generalized Jordan canonical form has a semisimple generalized Jordan block

**To prove**: The conjugacy class of in does not split in .

**Proof**:

Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|

1 | Without loss of generality, we can assume itself to be a generalized Jordan form matrix | (think about it) | |||

2 | The centralizer of in contains the internal direct product of the centralizers of each of the Jordan blocks. | (think about it) | |||

3 | The image of the centralizer of in under the determinant map contains the subgroup generated by the images (under the determinant map) of the centralizers of each of the Jordan blocks in their respective-sized matrix groups | Step (2) | (think about it) | ||

4 | If there is a semisimple generalized Jordan block, the image of its centralizer under the determinant map is all of . | Fact (2) | Follows because the norm map is the same as the determinant map (elaborate, link to additional fact eventually)
| ||

5 | The image of the centralizer of in under the determinant map is all of | has a semisimple generalized Jordan block | Steps (3), (4) | Step-combination direct | |

6 | The -conjugacy class of does not split in | Fact (1) | Step (5) | Step-fact combination direct |

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