General semilinear group of degree one

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Definition

Let K be a field. The general semilinear group of degree one over K, denoted \Gamma L(1,K), is defined as the general semilinear group of degree one over K. Explicitly, it is the external semidirect product:

\Gamma L (1,K) = GL(1,K) \rtimes \operatorname{Aut}(K) = K^\ast \rtimes \operatorname{Aut}(K)

where GL(1,K) = K^\ast is the multiplicative group of K, and \operatorname{Aut}(K) denotes the group of field automorphisms of K.

If k is the prime subfield of K, and K is a Galois extension of k (note that this case always occurs for K a finite field), then \operatorname{Aut}(K) = \operatorname{Gal}(K/k) and we get:

\Gamma L (1,K) = GL(1,K) \rtimes \operatorname{Gal}(K/k) = K^\ast \rtimes \operatorname{Gal}(K/k)

If K is a finite field of size q, this group is written as \Gamma L(1,q).

Particular cases

For a finite field

Suppose K is a finite field of size q, where q is a prime power with underlying prime p, so that q = p^r for a positive integer r. p is the characteristic of K. In this case, K^\ast is cyclic of order q - 1 (see multiplicative group of a finite field is cyclic) and \operatorname{Gal}(K/k) is cyclic of order r (generated by the Frobenius map a \mapsto a^p).

Thus, \Gamma L(1,K) is a metacyclic group of order r(q - 1) with presentation:

\langle a,x \mid a^q = a, x^r = e, xax^{-1} = a^p \rangle

(here e denotes the identity element).