# Dickson invariant

## Definition

### For an orthogonal group

Suppose is a field, is a natural number, and is a nondegenerate symmetric bilinear form for the -dimensional vector space . Let be the orthogonal group corresponding to .

The **Dickson invariant** is a group homomorphism from to cyclic group:Z2 (which we think of as a integers modulo 2) defined in the following equivalent ways:

- 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 .
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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.

### For a Clifford group

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### For a Pin group

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