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NB. Note that the notion of free group is different from the notion free abelian group.
The group (Z,+) of integers is free; we can take S = {1}. A free group on a two-element subset S occurs in the proof of the Banach-Tarski paradox and is described there.
If S is any set, there always exists a free group on S. This free group on S is essentially unique in the following sense: if F1 and F2 are two free groups on the set S, then F1 and F2 are isomorphic, and furthermore there exists precisely one group isomorphism f : F1 -> F2 such that f(s) = s for all s in S.
This free group on S is denoted by F(S) and can be constructed as follows. For every s in S, we introduce a new symbol s-1. We then form the set of all finite strings consisting of symbols of S and their inverses. Two such strings are considered equivalent if one arises from the other by replacing two adjacent symbols ss-1 or s-1s by the empty string. This generates an equivalence relation on the set of strings; its quotient set is defined to be F(S). Because the equivalence relation is compatible with string concatenation, F(S) becomes a group with string concatenation as operation.
If S is the empty set, then F(S) is the trivial group consisting only of its identity element.
The free group on S is characterized by the following universal property: if G is any group and
is any function, then there exists a unique group homomorphism
such that
for all s in S.
Free groups are thus instances of the more general concept of free objectThe idea of a free object in mathematics is one of the basics of abstract algebra. It is part of universal algebra, in the sense that it relates to all types of algebraic structure (with finitary operations); but on the other hand it has a clean formulatis in category theoryCategory theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. It is half-jokingly known as "generalized abstract nonsense". See list of category theory topics for a breakdown of relevan. Like most universal constructions, they give rise to a pair of adjoint functors.
Any group G is isomorphic to a quotient groupIn mathematics, given a group G and a normal subgroup N of G the quotient group or factor group of G over N is a group that intuitively "collapses" the normal subgroup N to the identity element. The quotient group is written G ''N and is usually spoken in of some free group F(S). If S can be chosen to be finite here, then G is called finitely generated.
If F is a free group on S and also on T, then S and T have the same cardinalityThe cardinality of a set is a property that describes the size of the set by describing it using a cardinal number. Sometimes we refer to this notion in a numerical way, so in the case of finite sets, the cardinality of the set is just the number of eleme. This cardinality is called the rank of the free group F. For every cardinal number k, there is, up toIn mathematics, the jargon term up to xxxx" is used to describe a situation in which members of an equivalence class can be regarded as a single entity for some purpose. xxxx" describes a property or process which transforms an element into one from the s isomorphism, exactly one free group of rank k.
If S has more than one element, then F(S) is not abelianAbstract algebra Algebra Group theory In mathematics, an abelian group is a commutative group, i. a group G ) such that a b b a for all a and b in G''. Abelian groups are named after Niels Henrik Abel. Notation There are two main notational conventions fo, and in fact the centerIn abstract algebra, the center (or centre of a group G is the set Z ''G of all elements in G which commute with all the elements of G''. Specifically, Z ''G z ∈ G | gz zg for all g ∈ G Note that Z ''G is a subgroup of G — if x and y are in Z '' of F(S) is trivial (that is, consists only of the identity element).
A free group of finite rank n > 1 has an exponential growth rate of order 2n − 1.
Any subgroup of free group is free. This statement easily follows from the following:
Any connected graph can be viewed as a path-connected topological space by treating an edge between two vertices as a continuous path between those vertices. With this understanding, the fundamental group of every connected graph is free and any free group is isomorphic to a fundamental group of a graph.
A free group of rank k clearly has subgroups of every rank less than k. Less obviously, a free group of rank greater than 1 has subgroups of all countable ranks.