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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 ...
Since the edge groups of Z are trivial, it follows that H is equal to the free product of the vertex groups of Z and the free group F(X) which is the fundamental group (in the standard topological sense) of the underlying graph Z of Z. This implies the conclusion of the Kurosh subgroup theorem.
The category-theoretical product in Grp is just the direct product of groups while the category-theoretical coproduct in Grp is the free product of groups. The zero objects in Grp are the trivial groups (consisting of just an identity element).
Product of group subsets; wreath product; free product; central product This page was last edited on 29 December 2020, at 00:45 (UTC). Text is available under the ...
Two free groups F S and F T are isomorphic if and only if S and T have the same cardinality. This cardinality is called the rank of the free group F. Thus for every cardinal number k, there is, up to isomorphism, exactly one free group of rank k. A free group of finite rank n > 1 has an exponential growth rate of order 2n − 1. A few other ...
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To see this, given a group G, consider the free group F G on G. By the universal property of free groups, there exists a unique group homomorphism φ : F G → G whose restriction to G is the identity map. Let K be the kernel of this homomorphism. Then K is normal in F G, therefore is equal to its normal closure, so G | K = F G /K.