For an orthogonal group
- The Dickson invariant of is the parity (i.e., value mod 2) of the rank of the linear transformation where is the identity transformation. In other words, if has even rank, the Dickson invariant is 0 mod 2 (the trivial element of the group of integers mod 2). If has odd rank, the Dickson invariant is 1 mod 2.
- In the case that is generated by reflections, the Dickson invariant is the parity of the number of reflections that need to be multiplied to get . PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
If the characteristic of is not 2, then the determinant map and Dickson invariant are equivalent in the following sense: the Dickson invariant is 0 mod 2 iff the determinant is 1 as an element of , and the Dickson invariant is 1 mod 2 iff the determinant is -1 as an element of .
If the characteristic of is 2, then the determinant of any element in any orthogonal group is always 1 as an element of , so the Dicksoon invariant carries more information than the determinant.