Class number formula - Revision history

en>Enyokoyama: /* Dirichlet class number formula */ add a link "Harold Davenport"

2014-04-23T10:31:46Z

Dirichlet class number formula: add a link "Harold Davenport"

← Older revision		Revision as of 12:31, 23 April 2014
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	~~{{multiple issues\|~~		Hello. Allow me introduce the author. Her title is Emilia Shroyer but it's not the most female name out there. South Dakota is where I've usually been residing. Doing ceramics is what love doing. I utilized to be unemployed but now I am a librarian and the wage has been really satisfying.<br><br>Here is my web page: [http://premium.asslikethat.com/blog/9567 at home std testing]
	~~{{expert-subject\|date=November 2008}}~~
	~~{{unreferenced\|date=November 2008}}~~
	}}

	~~In mathematical field of [[geometric group theory]], a '''length function''' is a function that assigns a number to each element of a group.~~

	~~==Definition==~~
	~~A '''length function''' ''L'' : ''G'' → '''R'''<sup>+</sup> on a [[group (mathematics)\|group]] ''G'' is a function satisfying:~~

	~~:<math>\begin{align}L(e) &= 0,\\~~
	~~L(g^{-1}) &= L(g)\\~~
	~~L(g_1 g_2) &\leq L(g_1) + L(g_2), \quad\forall g_1, g_2 \in G~~.
	~~\end{align}</math>~~

	~~Compare with~~ the ~~axioms for a [[Metric (mathematics)\|metric]] and a [[filtered algebra]]~~.

	~~==Word metric==~~
	~~{{main\|Word metric}}~~
	~~An important example of a length is the [[word metric]]: given a [[presentation of a group]] by generators and relations, the length of an element~~ is ~~the length of the shortest word expressing~~ it.

	~~[[Coxeter group]]~~s ~~(including~~ the ~~[[symmetric group]]) have combinatorial important length functions, using the simple reflections as generators (thus each simple reflection has length 1)~~. ~~See also: [[length of a Weyl group element]]~~.

	~~A [[longest element of a Coxeter group]]~~ is ~~both important and unique up to conjugation (up~~ to ~~different choice of simple reflections).~~

	~~==Properties==~~
	~~A group with~~ a ~~length function does ''not'' form a [[filtered group]], meaning that~~ the ~~[[sublevel set]]s~~ <~~math~~>~~S_i := \{g \mid L(g) \leq i\}~~<~~/math~~> ~~do not form subgroups in general.~~

	~~However, the~~ [~~[group ring\|group algebra]] of a group with a length functions forms a [[filtered algebra]]~~: ~~the axiom <math>L(gh) \leq L(g)+L(h)<~~/~~math> corresponds to the filtration axiom~~.

	~~{{PlanetMath attribution\|id=4365\|title=Length function}}~~

	~~[[Category:Group theory]]~~
	~~[[Category:Geometric group theory]~~]

en>Enyokoyama: /* Dirichlet class number formula */ link to Dirichlet charactor

2013-07-23T14:24:54Z

Dirichlet class number formula: link to Dirichlet charactor

← Older revision		Revision as of 16:24, 23 July 2013
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	~~Golda~~ is ~~what~~'~~s created~~ on ~~my beginning certification even though it~~ is ~~not~~ the ~~name on my birth certificate~~. ~~To climb~~ is ~~something she would~~ by ~~no means give~~ up. I'~~ve usually loved living in Kentucky but now I~~'~~m contemplating other choices. I am an invoicing officer and I~~'~~ll be promoted quickly.~~<br><br>~~Look into my homepage psychic readers -~~ [~~http~~:/~~/appin~~.~~co.kr/board_Zqtv22/688025 http~~:~~//appin.co.kr/board_Zqtv22/688025~~] -		{{multiple issues\|
			{{expert-subject\|date=November 2008}}
			{{unreferenced\|date=November 2008}}
			}}

			In mathematical field of [[geometric group theory]], a '''length function''' is a function that assigns a number to each element of a group.

			==Definition==
			A '''length function''' ''L'' : ''G'' → '''R'''<sup>+</sup> on a [[group (mathematics)\|group]] ''G'' is a function satisfying:

			:<math>\begin{align}L(e) &= 0,\\
			L(g^{-1}) &= L(g)\\
			L(g_1 g_2) &\leq L(g_1) + L(g_2), \quad\forall g_1, g_2 \in G.
			\end{align}</math>

			Compare with the axioms for a [[Metric (mathematics)\|metric]] and a [[filtered algebra]].

			==Word metric==
			{{main\|Word metric}}
			An important example of a length is the [[word metric]]: given a [[presentation of a group]] by generators and relations, the length of an element is the length of the shortest word expressing it.

			[[Coxeter group]]s (including the [[symmetric group]]) have combinatorial important length functions, using the simple reflections as generators (thus each simple reflection has length 1). See also: [[length of a Weyl group element]].

			A [[longest element of a Coxeter group]] is both important and unique up to conjugation (up to different choice of simple reflections).

			==Properties==
			A group with a length function does ''not'' form a [[filtered group]], meaning that the [[sublevel set]]s <math>S_i := \{g \mid L(g) \leq i\}</math> do not form subgroups in general.

			However, the [[group ring\|group algebra]] of a group with a length functions forms a [[filtered algebra]]: the axiom <math>L(gh) \leq L(g)+L(h)</math> corresponds to the filtration axiom.

			{{PlanetMath attribution\|id=4365\|title=Length function}}

			[[Category:Group theory]]
			[[Category:Geometric group theory]]

en>Bineapple: /* General statement of the class number formula */

2012-08-06T06:44:59Z

General statement of the class number formula

New page

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