Outer semilinear group

From Groupprops
Jump to: navigation, search

Definition

Suppose K is a field and n is a natural number. The outer semilinear group of degree n over K, denoted O\Gamma L(n,K), is defined as the external semidirect product:

GL(n,K) \rtimes (\operatorname{Aut}(K) \times \mathbb{Z}/2\mathbb{Z})

where \operatorname{Aut}(K) acts coordinate-wise on the matrix entries by automorphisms and the non-identity element of \mathbb{Z}/2\mathbb{Z} 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 K is a Galois extension of its prime subfield k (note that this is always true when K is a finite field), \operatorname{Aut}(K) = \operatorname{Gal}(K/k), so the group can also be written as:

GL(n,K) \rtimes (\operatorname{Gal}(K/k) \times \mathbb{Z}/2\mathbb{Z})