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  2. Free product - Wikipedia

    en.wikipedia.org/wiki/Free_product

    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 ...

  3. Normal form for free groups and free product of groups

    en.wikipedia.org/wiki/Normal_form_for_free...

    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 ...

  4. Product of groups - Wikipedia

    en.wikipedia.org/wiki/Product_of_groups

    In mathematics, a product of groups usually refers to a direct product of groups, ... Product of group subsets; wreath product; free product; central product

  5. Category of groups - Wikipedia

    en.wikipedia.org/wiki/Category_of_groups

    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).

  6. Kurosh subgroup theorem - Wikipedia

    en.wikipedia.org/wiki/Kurosh_subgroup_theorem

    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.

  7. Direct product of groups - Wikipedia

    en.wikipedia.org/wiki/Direct_product_of_groups

    In mathematics, specifically in group theory, the direct product is an operation that takes two groups G and H and constructs a new group, usually denoted G × H. This operation is the group-theoretic analogue of the Cartesian product of sets and is one of several important notions of direct product in mathematics.

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  9. Free group - Wikipedia

    en.wikipedia.org/wiki/Free_group

    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 ...