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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.
Free probability is a mathematical theory that studies non-commutative random variables. The "freeness" or free independence property is the analogue of the classical notion of independence, and it is connected with free products . This theory was initiated by Dan Voiculescu around 1986 in order to attack the free group factors isomorphism ...
In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subset such that every element of the group can be uniquely expressed as an integer combination of finitely ...
The 5th roots of unity in the complex plane form a group under multiplication. Each non-identity element generates the group. In abstract algebra, a generating set of a group is a subset of the group set such that every element of the group can be expressed as a combination (under the group operation) of finitely many elements of the subset and their inverses.
A presentation of a group determines a geometry, in the sense of geometric group theory: one has the Cayley graph, which has a metric, called the word metric. These are also two resulting orders, the weak order and the Bruhat order, and corresponding Hasse diagrams. An important example is in the Coxeter groups .
The Nielsen–Schreier theorem states that if H is a subgroup of a free group G, then H is itself isomorphic to a free group. That is, there exists a set S of elements which generate H, with no nontrivial relations among the elements of S . The Nielsen–Schreier formula, or Schreier index formula, quantifies the result in the case where the ...
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 ...
A free group has a unique normal form i.e. each element in is represented by a unique reduced word. Proof. An elementary transformation of a word w ∈ G {\displaystyle w\in G} consists of inserting or deleting a part of the form a a − 1 {\displaystyle aa^{-1}} with a ∈ S ± {\displaystyle a\in S^{\pm }} .