General semilinear group of degree one - Revision history

Vipul at 02:08, 1 June 2012

2012-06-01T02:08:28Z

← Older revision		Revision as of 02:08, 1 June 2012
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	Let <math>K</math> be a [[field]]. The '''general semilinear group of degree one''' over <math>K</math>, denoted <math>\Gamma L(1,K)</math>, is defined as the [[general semilinear group]] of degree one over <math>K</math>. Explicitly, it is the [[external semidirect product]]:		Let <math>K</math> be a [[field]]. The '''general semilinear group of degree one''' over <math>K</math>, denoted <math>\Gamma L(1,K)</math>, is defined as the [[general semilinear group]] of degree one over <math>K</math>. Explicitly, it is the [[external semidirect product]]:

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

			where <math>GL(1,K) = K^\ast</math> is the [[multiplicative group of a field\|multiplicative group]] of <math>K</math>, and <math>\operatorname{Aut}(K)</math> denotes the group of field automorphisms of <math>K</math>.

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

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

	where <math>GL(1,K) = K^\ast</math> is the [[multiplicative group of a field\|multiplicative group]] of <math>K</math>, <math>k</math> is the prime subfield of <math>K</math>, and <math>\operatorname{Gal}(K/k)</math> denotes the Galois group of <math>K</math> over <math>k</math>.

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

	==Particular cases==		==Particular cases==

Vipul at 15:58, 31 May 2012

2012-05-31T15:58:16Z

← Older revision		Revision as of 15:58, 31 May 2012
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	===For a finite field===		===For a finite field===

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

	~~Then~~, <math>\Gamma L(1,K)</math> is a metacyclic group of order <math>r(q - 1)</math> with presentation:		Thus, <math>\Gamma L(1,K)</math> is a metacyclic group of order <math>r(q - 1)</math> with presentation:

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

	(here <math>e</math> denotes the identity element).		(here <math>e</math> denotes the identity element).

Vipul at 15:52, 31 May 2012

2012-05-31T15:52:44Z

← Older revision		Revision as of 15:52, 31 May 2012
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	where <math>GL(1,K) = K^\ast</math> is the [[multiplicative group of a field\|multiplicative group]] of <math>K</math>, <math>k</math> is the prime subfield of <math>K</math>, and <math>\operatorname{Gal}(K/k)</math> denotes the Galois group of <math>K</math> over <math>k</math>.		where <math>GL(1,K) = K^\ast</math> is the [[multiplicative group of a field\|multiplicative group]] of <math>K</math>, <math>k</math> is the prime subfield of <math>K</math>, and <math>\operatorname{Gal}(K/k)</math> denotes the Galois group of <math>K</math> over <math>k</math>.

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

			==Particular cases==

			===For a finite field===

			Suppose <math>K</math> is a finite field of size <math>q</math>, where <math>q</math> is a [[prime power]] with underlying prime <math>p</math>, so that <math>q = p^r</math> for a positive integer <math>r</math>. <math>p</math> is the characteristic of <math>K</math>.

			Then, <math>\Gamma L(1,K)</math> is a metacyclic group of order <math>r(q - 1)</math> with presentation:

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

			(here <math>e</math> denotes the identity element).

Vipul: Created page with "==Definition== Let $K$ be a field. The '''general semilinear group of degree one''' over $K$ , denoted $\Gamma L(1,K)$ , is defined as the ..."

2012-05-31T15:50:27Z

Created page with "==Definition== Let <math>K</math> be a field. The '''general semilinear group of degree one''' over <math>K</math>, denoted <math>\Gamma L(1,K)</math>, is defined as the ..."

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==Definition==

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

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

where <math>GL(1,K) = K^\ast</math> is the [[multiplicative group of a field|multiplicative group]] of <math>K</math>, <math>k</math> is the prime subfield of <math>K</math>, and <math>\operatorname{Gal}(K/k)</math> denotes the Galois group of <math>K</math> over <math>k</math>.

General semilinear group of degree one - Revision history

Vipul at 02:08, 1 June 2012

Vipul at 15:58, 31 May 2012

Vipul at 15:52, 31 May 2012

Vipul: Created page with "==Definition== Let K be a field. The '''general semilinear group of degree one''' over K, denoted \Gamma L(1,K), is defined as the ..."

Vipul: Created page with "==Definition== Let $K$ be a field. The '''general semilinear group of degree one''' over $K$ , denoted $\Gamma L(1,K)$ , is defined as the ..."