General semiaffine group

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Definition

Suppose K is a field and n is a natural number. The general semiaffine group of degree n over K, denoted \Gamma A(n,K) or A \Gamma L(n,K), is defined in either of these equivalent ways:

  1. It is the group of all maps from K^n to K^n of the form v \mapsto A \sigma(v) + b where A \in GL(n,K), b \in K^n, \sigma \in \operatorname{Aut}(K).
  2. It is the semidirect product GA(n,K) \rtimes \operatorname{Aut}(K) of the general affine group GA(n,K) by the group \operatorname{Aut}(K) of field automorphisms of K, where the latter acts on the former by making the automorphism act coordinate-wise on the entries of the matrix and on the entries of the translation vector.
  3. It is the semidirect product K \rtimes \Gamma L(n,K) of the additive group of K by the general semilinear group \Gamma L(n,K) with the natural action of the latter on the former.

We can think of the group as an iterated semidirect product that can be associated two ways:

K \rtimes \Gamma L(n,K) = K \rtimes (GL(n,K) \rtimes \operatorname{Aut}(K)) = (K \rtimes GL(n,K)) \rtimes \operatorname{Aut}(K) = GA(n,K) \rtimes \operatorname{Aut}(K)

Suppose k is the prime subfield of K. Then, if K is a Galois extension of k, \operatorname{Aut}(K) is the Galois group \operatorname{Gal}(K/k). This case always occurs if K is a finite field.