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The free group FS is defined to be the group of all reduced words in S, with concatenation of words (followed by reduction if necessary) as group operation. The identity is the empty word. A reduced word is called cyclically reduced if its first and last letter are not inverse to each other.
Jun 5, 2020 · A free group in a variety of groups D D is defined analogously to a free group, but within D D. It is also called a D D - free group, or a relatively-free group (and also a reduced free group). If D D is defined by a set of identities v = 1 v = 1, where v ∈ V v ∈ V, then a free group of D D with a system of generators X X is isomorphic to ...
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free group on X can be deflned by a reduced group word on X. Moreover, difierent reduced words on X deflne difierent elements in G. We will say more about this in the next section. 1.2 Construction of a free group with basis X Let X be an arbitrary set. In this section we construct the canonical free group with basis X.
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May 24, 2024 · A group is called a free group if no relation exists between its group generators other than the relationship between an element and its inverse required as one of the defining properties of a group. For example, the additive group of integers is free with a single generator, namely 1 and its inverse, -1. An example of an element of the free group on two generators is ab^2a^(-1), which is not ...
The group F= F(X) is called the free group on set X. Note. The following properties hold for free groups: (1) If |X| ≥ 2 then the free group on Xis nonabelian. (2) Every element of a free group (except 1) has infinite order. (3) If X= {a}, then the free group on Xis the infinite cyclic group hai ∼= Z.
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Feb 24, 2016 · We can now define the free group. Let F(S) (the "free group on S ") be the set of all reduced words on T, and define a multiplication ⋅: F(S) × F(S) → F(S) by w ⋅ w ′ = r(w ∗ w ′), i.e. the reduced word obtained by reducing the concatenation of w and w ′. You can then prove that (F(S), ⋅) is a group. Share. Cite.
8. FREE GROUPS §8.1. Definition Consider a presentation with no relations or relators such as x 1, x 2, … , x n | . We call this the free group on the generators. Clearly free groups are infinite (except for the trivial case | where there are no generators. This is the trivial group, with one element).
