Presentation of a group

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In mathematics, one method of defining a group is by a presentation. One specifies a set S of generators so that every element of the group can be written as a product of some of these generators, and a set R of relations among those generators. We then say G has presentation

\langle S \mid R\rangle.

Informally, G has the above presentation if it is the "freest group" generated by S subject only to the relations R. Formally, the group G is said to have the above presentation if it is isomorphic to the quotient of a free group on S by the normal subgroup generated by the relations R.

As a simple example, the cyclic group of order n has the presentation

\langle a \mid a^n = e\rangle.

where e is the group identity. This may be written equivalently as

\langle a \mid a^n\rangle,

since terms that don't include an equals sign are taken to be equal to the group identity. Such terms are called relators, distinguishing them from the relations that include an equals sign.

Every group has a presentation, and in fact many different presentations; a presentation is often the most compact way of describing the structure of the group.

A closely related but different concept is that of an absolute presentation of a group.

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[edit] Background

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:

s_1^{a_1} s_2^{a_2} \ldots s_n^{a_n}

where each si is an element of S, and si ≠ si+1 for any i; and each ai is any non-zero integer. In less formal terms, the group consists of words in the generators and their inverses, subject only to canceling a generator with its inverse.

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 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.

However, we have, for example, r f r = f, r7 = r−1, etc.; so such products are not unique in D8. Each such product equivalence can be expressed as an equality to the identity; such as

r f r f = 1
r 8 = 1
f 2 = 1

Informally, we can consider these products on the left hand side as being elements of the free group F = <r,f>, and can consider the subgroup R of F which is generated by these strings; each of which would also be equivalent to 1 when considered as products in D8.

If we then let N be the subgroup of F generated by all conjugates x−1 R x of R, then N is a normal subgroup of F; and each element of N, when considered as a product in D8, will also evaluate to 1. Thus D8 is isomorphic to the quotient group F/N. We then say that D8 has presentation

\langle r, f \mid r^8 = f^2 = (rf)^2 = 1\rangle.

[edit] Formal definition

Let S be a set and let <S> be the free group on S. Let R be a set of words on S, so R naturally gives a subset of <S>. To form a group with presentation <S|R> the idea is take the smallest quotient of <S> so that each element of R gets identified with the identity element. Note that R may not be a subgroup, let alone a normal subgroup of <S>, so we cannot take a naive quotient by R. The solution is to take the normal closure N of R in <S>, which is defined as the smallest normal subgroup in <S> which contains R. The group <S|R> is then defined as the quotient group

\langle S \mid R \rangle = \langle S \rangle / N.

The elements of S are called the generators of <S|R> and the elements of R are called the relators. A group G is said to have the presentation <S|R> if G is isomorphic to <S|R>.

It is a common practice to write relators in the form x = y where x and y are words on S. What this means is that y^{-1}x \in R. This has the intuitive meaning that the images of x and y are supposed to be equal in the quotient group. Thus e.g. rn in the list of relators is equivalent with rn = 1. Another common shorthand is to write [x,y] for a commutator xyx − 1y − 1.

A presentation is said to be finitely generated if S is finite and finitely related if R is finite. If both are finite it is said to be a finite presentation. A group is finitely generated (respectively finitely related, finitely presented) if it has a presentation that is finitely generated (respectively finitely related, a finite presentation).

If S is indexed by a set I consisting of all the natural numbers \mathbb{N} or a finite subset of them, then it is easy to set up a simple one to one coding (or Gödel numbering) f:\langle S\rangle\rightarrow\mathbb{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. We can then call a subset U of <S>\ recursive (respectively recursively enumerable) if f(U) is recursive (respectively recursively enumerable). If S is indexed as above and R recursively enumerable, then the presentation is a recursive presentation and the corresponding group is recursively presented. This usage may seem odd, but it is possible to prove that if a group has a presentation with R recursively enumerable then it has another one with R recursive.

For a finite group G, the multiplication table provides a presentation. We take S to be the elements gi of G and R to be all words of the form g_ig_jg_k^{-1}, where g_ig_j=g_k\ is an entry in the multiplication table. A presentation can then be thought of as a generalization of a multiplication table.

Every finitely presented group is recursively presented, but there are recursively presented groups that cannot be finitely presented. However a remarkable 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. 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. Therefore there are finitely generated groups that cannot be recursively presented.

[edit] Examples

The following table lists some examples of presentations for commonly studied groups. Note that in each case there are many other presentations that are possible. The presentation listed is not necessarily the most efficient one possible.

Group Presentation Comments
the free group on S \langle S \mid \varnothing \rangle A free group is "free" in the sense that it is subject to no relations.
Cn, the cyclic group of order n \langle a \mid a^n \rangle
D2n, the dihedral group of order 2n \langle r, f \mid r^n, f^2, (rf)^2 \rangle Here r represents a rotation and f a reflection
D∞, the infinite dihedral group \langle r, f \mid f^2, (rf)^2 \rangle
Dicn, the dicyclic group \langle r, f \mid r^{2n} = 1, r^n = f^2, frf^{-1} = r^{-1} \rangle The quaternion group is a special case when n = 2
Z × Z \langle x, y \mid xy=yx \rangle
Zm × Zn \langle x, y \mid x^m=1, y^n=1, xy=yx \rangle
the free abelian group on S \langle S \mid R \rangle where R is the set of all commutators of elements of the free group on S
the symmetric group, Sn generators: \sigma_1, \ldots, \sigma_{n-1}
relations:
  • {\sigma_i}^2 = 1,
  • \sigma_i\sigma_j = \sigma_j\sigma_i \mbox{ if } j \neq i\pm 1,
  • σiσi + 1σi = σi + 1σiσi + 1

The last set of relations can be transformed into

  • (σiσi + 1)3 = 1

using {\sigma_i}^2=1 .

 Here σi is the permutation that swaps the ith element with the i+1 one. The product σiσi + 1 is a 3-cycle on the set {i,i + 1,i + 2}.
the braid group, Bn generators: \sigma_1, \ldots, \sigma_{n-1}

relations:

  • \sigma_i\sigma_j = \sigma_j\sigma_i \mbox{ if } j \neq i\pm 1,
  • σiσi + 1σi = σi + 1σiσi + 1
 Note the similarity with the symmetric group; the only difference is the removal of the relation {\sigma_i}^2 = 1.
the tetrahedral group, T ≅ A4 \langle s,t \mid s^2, t^3, (st)^3 \rangle
the octahedral group, O ≅ S4 \langle s,t \mid s^2, t^3, (st)^4 \rangle
the icosahedral group, I ≅ A5 \langle s,t \mid s^2, t^3, (st)^5 \rangle
the Quaternion group, Q \langle i,j \mid i^4, i^2j^2, ijij^{-1} \rangle
SL_2(\mathbb{Z}) \langle a,b \mid aba=bab, (aba)^4 \rangle topologically you can visualize a and b as Dehn twists on the torus
GL_2(\mathbb{Z}) \langle a,b,j \mid aba=bab, (aba)^4,j^2,(ja)^2,(jb)^2 \rangle
PSL_2(\mathbb{Z}) \langle a,b \mid a^2, b^3 \rangle PSL2(Z) is the free product of the cyclic groups Z2 and Z3
Heisenberg group \langle x,y,z \mid z=xyx^{-1}y^{-1}, xz=zx, yz=zy \rangle
Baumslag-Solitar group, B(m,n) \langle a, b \mid a^n = b a^m b^{-1} \rangle

[edit] Some theorems

Every group G has a presentation. To see this consider the free group <G> on G. Since G clearly generates itself one should be able to obtain it by a quotient of <G>. Indeed, by the universal property of free groups there exists a unique group homomorphism φ : <G> → G which covers the identity map. Let K be the kernel of this homomorphism. Then G clearly has the presentation <G|K>. Note that this presentation is highly inefficient as both G and K are much larger than necessary.

Every finite group has a finite presentation.

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.

[edit] Free product

If G has presentation <S|R> and H has presentation <T|Q> with S and T being disjoint then the free product G * H has presentation <S,T|R,Q>.

[edit] Direct product

If G has presentation <S|R> and H has presentation <T|Q> with S and T being disjoint then the direct product of G and H has presentation <S,T|R,Q, [S,T]>. Here [S,T] means that every element from S commutes with every element from T.

[edit] See also

[edit] References

  • Johnson, D. L. (1990). Presentations of Groups. Cambridge: Cambridge University Press. ISBN 0-521-37824-9.  Schreier's method, Nielsen's method, free presentations, subgroups and HNN extensions, Golod-Shafarevich theorem, etc.
  • Coxeter, H. S. M. and Moser, W. O. J. (1980). Generators and Relations for Discrete Groups. New York: Springer-Verlag. ISBN 0-387-09212-9.  This useful reference has tables of presentations of all small finite groups, the reflection groups, and so forth.