 The Cayley graph of the free group on two generators a and b In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses (disregarding trivial variations such as st-1 = su-1ut-1). Download high resolution version (1996x1996, 57 KB) Wikipedia does not have an article with this exact name. ...
Download high resolution version (1996x1996, 57 KB) Wikipedia does not have an article with this exact name. ...
The Cayley graph of the free group on two generators a and b In mathematics, a Cayley graph, named after Arthur Cayley, is a graph that encodes the structure of a group. ...
Main article: History of mathematics The evolution of mathematics can be seen to be an ever increasing series of abstractions. ...
In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...
A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes) X; Y ⊇ X...
Note that the notion of free group is different from the notion free abelian group. In abstract algebra, a free abelian group is an abelian group that has a basis in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. ...
Examples
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. The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ...
The Banach-Tarski paradox: A ball can be decomposed and reassembled into two balls the same size as the original. ...
Construction 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. In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. ...
In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. ...
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. Generally, string is a thin piece of fiber which is used to tie, bind, or hang other objects. ...
In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ...
In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x in X | x ~ a } The notion of equivalence classes is useful for constructing sets...
If S is the empty set, then F(S) is the trivial group consisting only of its identity element. In mathematics, the empty set is the set with no elements. ...
In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ...
Universal property The free group on S is characterized by the following universal property: if G is any group and In various branches of mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ...
- f : S → G
is any function, then there exists a unique group homomorphism In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ...
Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element...
- T : F(S) → G
such that - T(s) = f(s)
for all s in S.
Free groups are thus instances of the more general concept of free objects in category theory. Like most universal constructions, they give rise to a pair of adjoint functors. The idea of a free object in mathematics is one of the basics of abstract algebra. ...
Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ...
The existence of many pairs of adjoint functors is a major observation of the branch of mathematics known as category theory. ...
Facts and theorems Any group G is isomorphic to a quotient group of some free group F(S). If S can be chosen to be finite here, then G is called finitely generated. In 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. ...
If F is a free group on S and also on T, then S and T have the same cardinality. This cardinality is called the rank of the free group F. For every cardinal number k, there is, up to isomorphism, exactly one free group of rank k. In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers. ...
Look up Up to in Wiktionary, the free dictionary Modern Slang In modern slang, up to means you are either willing to engage in an act (Sally is up to going to the park), capable of an act (Im sorry, Im just not up to it) or are...
If S has more than one element, then F(S) is not abelian, and in fact the center of F(S) is trivial (that is, consists only of the identity element). In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ...
In 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...
A free group of finite rank n > 1 has an exponential growth rate of order 2n − 1. In mathematics, a quantity that grows exponentially is one that grows at a rate proportional to its size. ...
In group theory, the growth rate of a group with respect to a symmetric generating set is a notion that describes how fast a group grows. ...
Nielsen-Schreier theorem: Any subgroup of a free group is free. In mathematics, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group operation...
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. In mathematics the term countable set is used to describe the size of a set, e. ...
Tarski's Problems Around 1945, Alfred Tarski asked whether the free groups on two or more generators have the same first order theory, and whether this theory is decidable. Independently, a proof for both problems, and a proof of the first problem, have been announced (both in the affirmative). Neither has yet been judged correct and complete. For details, see the open problems at [1]. Alfred Tarski (January 14, 1901 in Warsaw–October 26, 1983 in Berkeley, USA) was a Polish logician considered to be one of the greatest logicians of all time in a manner after Aristotle, Gottlob Frege, and Kurt Gödel. ...
In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the models which underlie mathematical systems. ...
A logical system is decidable iff there exists an algorithm such that for every well-formed formula in that system there exists a maximum finite number N of steps such that the algorithm is capable of deciding in less than or equal to N algorithmic steps whether the formula is...
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