General semiaffine group of degree one

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Definition

The general semiaffine group of degree one over a field K, denoted \Gamma A(1,K) or A\Gamma L(1,K), is the general semiaffine group of degree one over K. It can be defined explicitly as the group of all permutations of K that can be expressed in the form:

x \mapsto a \sigma(x) + b, a \in K^\ast, b \in K, \sigma \in \operatorname{Aut}(K)

It can be viewed as an iterated semidirect product:

(K \rtimes K^\ast) \rtimes \operatorname{Aut}(K) = K \rtimes (K^\ast \rtimes \operatorname{Aut}(K))

The left parenthesized expression shows that the group can be viewed as a semidirect product with base the general affine group of degree one and acting group the Galois group:

\Gamma A(1,K) = GA(1,K) \rtimes \operatorname{Aut}(K)

The right parenthesized expression shows that the group can be viewed as a semidirect product with base the additive group of the field and acting group the general semilinear group of degree one:

\Gamma A(1,K) = K \rtimes \Gamma L(1,K)

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

Alternative definition as automorphisms of a polynomial ring

The group \Gamma A(1,K) can also be defined as the group of all ring automorphisms of the polynomial ring K[x]. The subgroup \operatorname{Aut}_K(K[x]) of those automorphisms that fix the base field K can be identified with the general affine group of degree one GA(1,K).