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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 “universal” group having these properties, in the sense that any two homomorphisms from G and H into a group K factor uniquely through a ...
In mathematics, particularly in combinatorial group theory, a normal form for a free group over a set of generators or for a free product of groups is a representation of an element by a simpler element, the element being either in the free group or free products of group. In case of free group these simpler elements are reduced words and in ...
G ∗ H = 〈 S G ∪ S H | R G ∪ R H 〉. Unlike the direct product, elements of the free product cannot be represented by ordered pairs. In fact, the free product of any two nontrivial groups is infinite. The free product is actually the coproduct in the category of groups.
The free group G = π 1 (X) has n = 2 generators corresponding to loops a,b from the base point P in X.The subgroup H of even-length words, with index e = [G : H] = 2, corresponds to the covering graph Y with two vertices corresponding to the cosets H and H' = aH = bH = a −1 H = b − 1 H, and two lifted edges for each of the original loop-edges a,b.
A free group of finite rank n > 1 has an exponential growth rate of order 2n − 1. A few other related results are: The Nielsen–Schreier theorem: Every subgroup of a free group is free. Furthermore, if the free group F has rank n and the subgroup H has index e in F, then H is free of rank 1 + e(n–1).
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Product of group subsets; wreath product; free product; central product This page was last edited on 29 ... Code of Conduct; Developers; Statistics; Cookie statement;
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