en>Yobot: WP:CHECKWIKI error fixes using AWB (10616)

2015-01-01T09:44:24Z

WP:CHECKWIKI error fixes using AWB (10616)

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en>JFB80: /* Definition */ It is elementary calculus and doesnt need an obscure reference.

2014-02-19T11:21:23Z

Definition: It is elementary calculus and doesnt need an obscure reference.

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5.65.147.180: /* Uses in mathematical statistics */

2013-12-07T22:19:18Z

Uses in mathematical statistics

← Older revision		Revision as of 00:19, 8 December 2013
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	~~Historical past~~ of the of the ~~author~~ is ~~generally Gabrielle Lattimer~~. ~~For years she~~'s ~~been working even though~~ a ~~library assistant~~. ~~For~~ a ~~while she~~'s ~~yet been~~ in ~~[https:~~//~~www~~.~~google~~.~~com~~/~~search?hl~~=~~en&gl~~=~~us&tbm~~=~~nws&q~~=~~Massachusetts&btnI~~=~~lucky Massachusetts~~]~~. As~~ a ~~woman what your woman really likes~~ is ~~[http~~:/~~/Www~~.~~tumblr~~.~~com~~/~~tagged~~/~~mah+jongg mah jongg] but she hasn~~'~~t ever made~~ a ~~dime to barefoot jogging~~. ~~She~~ are ~~running and maintaining~~ a ~~meaningful blog here: http~~://~~circuspartypanama~~.~~com~~<br><br>~~Here~~ is ~~my web~~-~~site .~~.. [~~http~~://~~circuspartypanama~~.~~com hack clash~~ of ~~clans android~~]		In [[mathematics]], specifically [[group theory]], the '''free product''' is an operation that takes two [[group (mathematics)\|groups]] ''G'' and ''H'' and constructs a new group ''G'' ∗ ''H''. The result contains both ''G'' and ''H'' as [[subgroup]]s, is [[Generating set of a group\|generated]] by the elements of these subgroups, and is the “most general” group having these properties. Unless one of the groups ''G'' and ''H'' is trivial, the free product is always infinite. The construction of a free product is similar in spirit to the construction of a [[free group]] (the most general group that can be made from a given set of generators).

			The free product is the [[coproduct]] in the [[category of groups]]. That is, the free product plays the same role in group theory that [[disjoint union]] plays in [[set theory]], or that the [[direct sum of modules\|direct sum]] plays in [[module theory]]. Even if the groups are commutative, their free product is not, unless one of the two groups is the [[trivial group]]. Therefore the free product is not the coproduct in the [[category of abelian groups]].

			The free product is important in [[algebraic topology]] because of [[van Kampen's theorem]], which states that the [[fundamental group]] of the [[union (set theory)\|union]] of two [[Connected space\|path-connected]] [[topological space]]s is always an '''amalgamated free product''' of the fundamental groups of the spaces. In particular, the fundamental group of the [[wedge sum]] of two spaces (i.e. the space obtained by joining two spaces together at a single point) is simply the free product of the fundamental groups of the spaces.

			Free products are also important in [[Bass–Serre theory]], the study of groups [[Group action\|acting]] by automorphisms on [[Tree (graph theory)\|trees]]. Specifically, any group acting with finite vertex stabilizers on a tree may be constructed from finite groups using amalgamated free products and [[HNN extension]]s. Using the action of the [[modular group]] on a certain [[tessellation]] of the [[hyperbolic geometry\|hyperbolic plane]], it follows from this theory that the modular group is [[isomorphic]] to the free product of [[cyclic group]]s of orders 4 and 6 amalgamated over a cyclic group of order 2.

			==Construction==
			If ''G'' and ''H'' are groups, a '''[[word (group theory)\|word]]''' in ''G'' and ''H'' is a product of the form
			:<math>s_1 s_2 \cdots s_n,</math>
			where each ''s''<sub>''i''</sub> is either an element of ''G'' or an element of ''H''. Such a word may be '''reduced''' using the following operations:
			* Remove an instance of the identity element (of either ''G'' or ''H'').
			* Replace a pair of the form ''g''<sub>1</sub>''g''<sub>2</sub> by its product in ''G'', or a pair ''h''<sub>1</sub>''h''<sub>2</sub> by its product in ''H''.
			Every reduced word is an alternating product of elements of ''G'' and elements of ''H'', e.g.
			:<math>g_1 h_1 g_2 h_2 \cdots g_k h_k.</math>
			The '''free product''' ''G'' ∗ ''H'' is the group whose elements are the reduced words in ''G'' and ''H'', under the operation of concatenation followed by reduction.

			For example, if ''G'' is the infinite cyclic group <''x''>, and ''H'' is the infinite cyclic group <''y''>, then every element of ''G'' ∗ ''H'' is an alternating product of powers of ''x'' with powers of ''y''. In this case, ''G'' ∗ ''H'' is isomorphic to the free group generated by ''x'' and ''y''.

			==Presentation==
			Suppose that
			:<math>G = \langle R_G \mid S_G \rangle</math>
			is a [[Presentation of a group\|presentation]] for ''G'' (where ''R''<sub>''G''</sub> is a set of generators and ''S''<sub>''G''</sub> is a set of relations), and suppose that
			:<math>H = \langle R_H \mid S_H \rangle</math>
			is a presentation for ''H''. Then
			:<math>G * H = \langle R_G \cup R_H \mid S_G \cup S_H \rangle.</math>
			That is, ''G'' ∗ ''H'' is generated by the generators for ''G'' together with the generators for ''H'', with relations consisting of the relations from ''G'' together with the relations from ''H'' (assume here no notational clashes so that these are in fact [[disjoint union]]s).

			For example, suppose that ''G'' is a cyclic group of order 4,
			:<math>G = \langle x \mid x^4 = 1 \rangle,</math>
			and ''H'' is a cyclic group of order 5
			:<math>H = \langle y \mid y^5 = 1 \rangle.</math>
			Then ''G'' ∗ ''H'' is the infinite group
			:<math>G * H = \langle x, y \mid x^4 = y^5 = 1 \rangle.</math>

			Because there are no relations in a free group, the free product of free groups is always a free group. In particular,
			:<math>F_m * F_n \cong F_{m+n},</math>
			where ''F''<sub>''n''</sub> denotes the free group on ''n'' generators.

			==Generalization: Free product with amalgamation==
			The more general construction of '''free product with amalgamation''' is correspondingly a [[pushout (category theory)\|pushout]] in the same [[category theory\|category]]. Suppose ''G'' and ''H'' are given as before, along with [[group homomorphism]]s

			:<math>\varphi : F \rightarrow G\mbox{ and }\psi : F \rightarrow H.</math>

			where ''F'' is some arbitrary group. Start with the free product ''G'' ∗ ''H'' and adjoin as relations

			:<math>\varphi(f)\psi(f)^{-1}=1</math>

			for every ''f'' in ''F''. In other words take the [[Conjugate closure\|smallest normal subgroup]] ''N'' of ''G'' ∗ ''H'' containing all elements on the [[Left-hand side and right-hand side of an equation\|left-hand side]] of the above equation, which are tacitly being considered in ''G'' ∗ ''H'' by means of the inclusions of ''G'' and ''H'' in their free product. The free product with amalgamation of ''G'' and ''H'', with respect to φ and ψ, is the [[quotient group]]

			:<math>(G * H)/N.\,</math>

			The amalgamation has forced an identification between φ(''F'') in ''G'' with ψ(''F'') in ''H'', element by element. This is the construction needed to compute the fundamental group of two connected spaces joined along a connected subspace, with ''F'' taking the role of the fundamental group of the subspace. See: [[Seifert–van Kampen theorem]].

			Free products with amalgamation and a closely related notion of [[HNN extension]] are basic building blocks in Bass–Serre theory of groups acting on trees.

			==In other branches==
			One may similarly define free products of other algebraic structures than groups, including [[algebra over a field\|algebras over a field]]. Free products of algebras of [[random variable]]s play the same role in defining "[[free independence\|freeness]]" in the theory of [[free probability]] that [[Cartesian product]]s play in defining [[statistical independence]] in classical [[probability theory]].

			==See also==
			*[[Direct product of groups]]
			*[[Free object]]
			*[[Graph of groups]]
			*[[Kurosh subgroup theorem]]
			*[[Universal property]]

			==References==
			* {{planetmath reference\|id=6574\|title=Free product}}
			* {{planetmath reference\|id=3944\|title=Free product with amalgamated subgroup}}

			==Notes==
			{{Reflist}}

			[[Category:Group theory]]
			[[Category:Algebraic topology]]
			[[Category:Free algebraic structures]]

63.225.85.69: /* Some properties of the cumulant generating function */ sign (!)

2012-07-13T06:11:52Z

Some properties of the cumulant generating function: sign (!)

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Historical past of the of the author is generally Gabrielle Lattimer. For years she's been working even though a library assistant. For a while she's yet been in [https://www.google.com/search?hl=en&gl=us&tbm=nws&q=Massachusetts&btnI=lucky Massachusetts]. As a woman what your woman really likes is [http://Www.tumblr.com/tagged/mah+jongg mah jongg] but she hasn't ever made a dime to barefoot jogging. She are running and maintaining a meaningful blog here: http://circuspartypanama.com<br><br>Here is my web-site ... [http://circuspartypanama.com hack clash of clans android]

Cumulant - Revision history

en>Yobot: WP:CHECKWIKI error fixes using AWB (10616)

en>JFB80: /* Definition */ It is elementary calculus and doesnt need an obscure reference.

5.65.147.180: /* Uses in mathematical statistics */

63.225.85.69: /* Some properties of the cumulant generating function */ sign (!)