Presentation of a group
A closely related but different concept is that of an absolute presentation of a group.A free group on a set S is a group where each element can be uniquely described as a finite length product of the form: where the si are elements of S, adjacent si are distinct, and ai are non-zero integers (but n may be zero).If G is any group, and S is a generating subset of G, then every element of G is also of the above form; but in general, these products will not uniquely describe an element of G. For example, the dihedral group D8 of order sixteen can be generated by a rotation, r, of order 8; and a flip, f, of order 2; and certainly any element of D8 is a product of r's and f's.Each such product equivalence can be expressed as an equality to the identity, such as Informally, we can consider these products on the left hand side as being elements of the free group F = ⟨r, f⟩, and let R = ⟨rfrf, r8, f2⟩.That is, we let R be the subgroup generated by the strings rfrf, r8, f2, each of which is also equivalent to 1 when considered as products in D8.It follows that each element of N, when considered as a product in D8, will also evaluate to 1; and thus that N is a normal subgroup of F. Thus D8 is isomorphic to the quotient group F/N.We often see R abbreviated, giving the presentation An even shorter form drops the equality and identity signs, to list just the set of relators, which is {r 8, f 2, (rf )2}.Although the notation ⟨S | R⟩ used in this article for a presentation is now the most common, earlier writers used different variations on the same format.is then defined as the quotient group The elements of S are called the generators ofThis has the intuitive meaning that the images of x and y are supposed to be equal in the quotient group.The definition of group presentation may alternatively be recast in terms of equivalence classes of words on the alphabet[1] This point of view is particularly common in the field of combinatorial group theory.If S is indexed by a set I consisting of all the natural numbers N or a finite subset of them, then it is easy to set up a simple one to one coding (or Gödel numbering) f : FS → N from the free group on S to the natural numbers, such that we can find algorithms that, given f(w), calculate w, and vice versa.However a theorem of Graham Higman states that a finitely generated group has a recursive presentation if and only if it can be embedded in a finitely presented group.[2] From this we can deduce that there are (up to isomorphism) only countably many finitely generated recursively presented groups.Bernhard Neumann has shown that there are uncountably many non-isomorphic two generator groups.One of the earliest presentations of a group by generators and relations was given by the Irish mathematician William Rowan Hamilton in 1856, in his icosian calculus – a presentation of the icosahedral group.[3] The first systematic study was given by Walther von Dyck, student of Felix Klein, in the early 1880s, laying the foundations for combinatorial group theory.[4] The following table lists some examples of presentations for commonly studied groups.The negative solution to the word problem for groups states that there is a finite presentation ⟨S | R⟩ for which there is no algorithm which, given two words u, v, decides whether u and v describe the same element in the group.This was shown by Pyotr Novikov in 1955[5] and a different proof was obtained by William Boone in 1958.[8] 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.Further, some properties of this graph (the coarse geometry) are intrinsic, meaning independent of choice of generators.