# Special unitary group

## Definition

Suppose is a separable quadratic extension of a field and is the unique automorphism of that fixes pointwise. The **special unitary group** of degree for this quadratic extension, denoted (if the extension being referred to is understood) is defined as the subgroup of the special linear group comprising those matrices on which the transpose-inverse map gives the same result as the entry-wise application of .

Here, is the matrix obtained by applying to each of the entries of .

Alternatively, we can define it as the intersection of the unitary group and the special linear group, both viewed as subgroups of the general linear group:

### For the real and complex numbers

The most typical usage of the term *special unitary group* is in the context where is the field of real numbers, is the field of complex numbers, and the automorphism is complex conjugation. In this case, the group is the subgroup of the special linear group comprising those matrices whose complex conjugate equals the transpose-inverse. When it's understood that we are working over the complex numbers, this group is sometimes just denoted .

### For a finite field

If is the (unique up to isomorphism) finite field of size a prime power , there is a unique quadratic extension of , and this extension is separable. The extension field is the finite field (unique up to isomorphism) of order . The automorphism is the map . The special unitary group for this extension may be denoted (the more standard choice) or (a less standard choice). Note that due to the ambiguity of notation, it is important to understand from context what exactly is meant.

Note that, if we denote this group by , then, somewhat confusingly: