# Outer semilinear group

From Groupprops

## Definition

Suppose is a field and is a natural number. The **outer semilinear group** of degree over , denoted , is defined as the external semidirect product:

where acts coordinate-wise on the matrix entries by automorphisms and the non-identity element of acts by the transpose-inverse map. Note that the two actions commute with each other, so we can combine these to get the action of the external direct product.

In the case that is a Galois extension of its prime subfield (note that this is always true when is a finite field), , so the group can also be written as: