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- Free_product abstract "In mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new group G ∗ H. The result contains both G and H as subgroups, is 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 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 of two path-connected topological spaces 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 acting by automorphisms on trees. Specifically, any group acting with finite vertex stabilizers on a tree may be constructed from finite groups using amalgamated free products and HNN extensions. Using the action of the modular group on a certain tessellation of the hyperbolic plane, it follows from this theory that the modular group is isomorphic to the free product of cyclic groups of orders 4 and 6 amalgamated over a cyclic group of order 2.".
- Free_product wikiPageID "360036".
- Free_product wikiPageRevisionID "571749298".
- Free_product hasPhotoCollection Free_product.
- Free_product id "3944".
- Free_product id "6574".
- Free_product title "Free product with amalgamated subgroup".
- Free_product title "Free product".
- Free_product subject Category:Algebraic_topology.
- Free_product subject Category:Free_algebraic_structures.
- Free_product subject Category:Group_theory.
- Free_product comment "In mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new group G ∗ H. The result contains both G and H as subgroups, is 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.".
- Free_product label "Free product".
- Free_product label "Freies Produkt".
- Free_product label "Prodotto libero".
- Free_product label "Producto libre de grupos".
- Free_product label "Produit libre".
- Free_product label "Свободное произведение".
- Free_product label "自由積".
- Free_product sameAs Freies_Produkt.
- Free_product sameAs Producto_libre_de_grupos.
- Free_product sameAs Produit_libre.
- Free_product sameAs Prodotto_libero.
- Free_product sameAs m.01_bcn.
- Free_product sameAs Q1454165.
- Free_product sameAs Q1454165.
- Free_product wasDerivedFrom Free_product?oldid=571749298.
- Free_product isPrimaryTopicOf Free_product.