Diagram showing the Cayley graph for the free group on two generators. Each vertex represents an element of the free group, and each edge represents multiplication by a or b.
In mathematics, the free groupFS over a given set S consists of all words that can be built from members of S, considering two words to be different unless their equality follows from the group axioms (e.g. st = suu−1t but s ≠ t−1 for s,t,u ∈ S). The members of S are called generators of FS, and the number of generators is the rank of the free group.
An arbitrary group G is called free if it is isomorphic to FS for some subset S of G, that is, if there is a subset S of G such that every element of G can be written in exactly one way as a product of finitely many elements of S and their inverses (disregarding trivial variations such as st = suu−1t).
A related but different notion is a free abelian group; both notions are particular instances of a free object from universal algebra. As such, free groups are defined by their universal property.
In mathematics, the freegroup FS over a given set S consists of all words that can be built from members of S, considering two words to be different unless...
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...
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...
any group on itself by left multiplication is free. This observation implies Cayley's theorem that any group can be embedded in a symmetric group (which...
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and it is connected with free products. This theory was initiated by Dan Voiculescu around 1986 in order to attack the freegroup factors isomorphism problem...
In mathematical group theory, the automorphism group of a freegroup is a discrete group of automorphisms of a freegroup. The quotient by the inner automorphisms...
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particularly in combinatorial group theory, a normal form for a freegroup over a set of generators or for a free product of groups is a representation of an...
mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not...
More generally, the fundamental group of a bouquet of r circles is the freegroup on r letters. The fundamental group of a wedge sum of two path connected...
cohomology groups are trivial. Given a set S {\displaystyle S} the associated freegroup G = Free ( S ) = ∗ s ∈ S Z {\displaystyle G={\text{Free}}(S)={\underset...
fundamental groups. For example, one can show that every subgroup of a freegroup is free. There are several natural questions arising from giving a group by its...
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subgroup of a finitely generated group need not be finitely generated. For example, let G {\displaystyle G} be the freegroup in two generators, x {\displaystyle...
DFS opened its first downtown duty-free store in Hainan Island, China, in partnership with Shenzhen Duty FreeGroup. Within a few months, DFS also announced...
sense if they are conjugate in the sense of group theory as elements of the freegroup generated by A. A free monoid is equidivisible: if the equation mn...
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