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| 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).
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| 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]].
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| 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.
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| 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.
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| ==Construction==
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| If ''G'' and ''H'' are groups, a '''[[word (group theory)|word]]''' in ''G'' and ''H'' is a product of the form
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| :<math>s_1 s_2 \cdots s_n,</math>
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| 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:
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| * Remove an instance of the identity element (of either ''G'' or ''H'').
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| * 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''.
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| Every reduced word is an alternating product of elements of ''G'' and elements of ''H'', e.g.
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| :<math>g_1 h_1 g_2 h_2 \cdots g_k h_k.</math>
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| 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.
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| 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''.
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| ==Presentation==
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| Suppose that
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| :<math>G = \langle R_G \mid S_G \rangle</math>
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| 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
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| :<math>H = \langle R_H \mid S_H \rangle</math>
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| is a presentation for ''H''. Then
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| :<math>G * H = \langle R_G \cup R_H \mid S_G \cup S_H \rangle.</math>
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| 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).
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| For example, suppose that ''G'' is a cyclic group of order 4,
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| :<math>G = \langle x \mid x^4 = 1 \rangle,</math>
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| and ''H'' is a cyclic group of order 5 | |
| :<math>H = \langle y \mid y^5 = 1 \rangle.</math>
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| Then ''G'' ∗ ''H'' is the infinite group
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| :<math>G * H = \langle x, y \mid x^4 = y^5 = 1 \rangle.</math>
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| Because there are no relations in a free group, the free product of free groups is always a free group. In particular,
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| :<math>F_m * F_n \cong F_{m+n},</math>
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| where ''F''<sub>''n''</sub> denotes the free group on ''n'' generators.
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| ==Generalization: Free product with amalgamation==
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| 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 | |
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| :<math>\varphi : F \rightarrow G\mbox{ and }\psi : F \rightarrow H.</math>
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| where ''F'' is some arbitrary group. Start with the free product ''G'' ∗ ''H'' and adjoin as relations
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| :<math>\varphi(f)\psi(f)^{-1}=1</math>
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| 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]]
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| :<math>(G * H)/N.\,</math> | |
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| 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]].
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| 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.
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| ==In other branches==
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| 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]].
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| ==See also==
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| *[[Direct product of groups]]
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| *[[Free object]]
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| *[[Graph of groups]]
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| *[[Kurosh subgroup theorem]]
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| *[[Universal property]]
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| ==References==
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| * {{planetmath reference|id=6574|title=Free product}}
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| * {{planetmath reference|id=3944|title=Free product with amalgamated subgroup}}
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| ==Notes==
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| {{Reflist}}
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| [[Category:Group theory]]
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| [[Category:Algebraic topology]]
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| [[Category:Free algebraic structures]]
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Most of the amend delivers a information of notable enhancements, the alignment of which could be the new Dynasty Struggle Manner. In specific mode, you can making claims combating dynasties and lessen utter rewards aloft beat.
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Consumers are a group towards coders that loves into play Cof. We are going to are continuously developing Hacks to speed up Levelling easily and to be more gems for clear. Without our hacks it does take you ages to assist you to reach your level.
Computing games are a lot of fun, but these items could be very tricky, also. If your organization are put on an actual game, go on the exact web and also appear for cheats. All games have some sorts of cheat or tricks that can make persons a lot easier. Only search in very own favorite search engine and consequently you can certainly discover cheats to get your action better.
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