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

Informally, G has the above presentation if it is the "free-est 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

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

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.

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:

where the s_i are distinct elements of S, and a_i are non-zero integers. 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 D 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 D is a product of rs and fs.

However, we have, for example, r f r = f, r ⁷ = r ⁻¹, etc.; so such products are not unique in D. 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 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 D.

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 D, will also evaluate to 1. Thus D is isomorphic to the quotient group F /N. We then say that D has presentation

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
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 $y^\{-1\}x\; \backslash in\; R$. This has the intuitive meaning that the images of x and y are supposed to be equal in the quotient group. Thus e.g. $r^n$ in the list of relators is equivalent with $r^n=1$.
Another common shorthand is to write $[x,y]$ for a commutator $xyx^\{-1\}y^\{-1\}$.

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 $\backslash mathbb\{N\}$ or a finite subset of them, then it is easy to set up a simple one to one coding (or Gödel numbering) $f:\backslash langle\; S\backslash rangle\backslash rightarrow\backslash mathbb\{N\}$ 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 $\backslash langle\; S\backslash rangle$ 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 $g\_i$ of $G$ and $R$ to be all words of the form $g\_ig\_jg\_k^\{-1\}$, where $g\_ig\_j=g\_k\backslash$ 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 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.

Examples

History

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.

Common 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.
{| border=1
!Group || Presentation || Comments
|-
| the free group on S || $\backslash langle\; S\; \backslash mid\; \backslash varnothing\; \backslash rangle$
| A free group is "free" in the sense that it is subject to no relations.
|-
| C_n, the cyclic group of order n
| $\backslash langle\; a\; \backslash mid\; a^n\; \backslash rangle\backslash ,\backslash !$
|
|-
| D_2n, the dihedral group of order 2n
| $\backslash langle\; r,\; f\; \backslash mid\; r^n,\; f^2,\; (rf)^2\; \backslash rangle\backslash ,\backslash !$
| Here r represents a rotation and f a reflection
|-
| D_∞, the infinite dihedral group
| $\backslash langle\; r,\; f\; \backslash mid\; f^2,\; (rf)^2\; \backslash rangle\backslash ,\backslash !$
|
|-
| Dic_n, the dicyclic group
| $\backslash langle\; r,\; f\; \backslash mid\; r^\{2n\}\; =\; 1,\; r^n\; =\; f^2,\; frf^\{-1\}\; =\; r^\{-1\}\; \backslash rangle\backslash ,\backslash !$
| The quaternion group is a special case when n = 2
|-
| Z × Z
| $\backslash langle\; x,\; y\; \backslash mid\; xy=yx\; \backslash rangle\backslash ,\backslash !$
|
|-
| Z_m × Z_n
| $\backslash langle\; x,\; y\; \backslash mid\; x^m=1,\; y^n=1,\; xy=yx\; \backslash rangle\backslash ,\backslash !$
|
|-
| the free abelian group on S
| $\backslash langle\; S\; \backslash mid\; R\; \backslash rangle\backslash ,\backslash !$ where R is the set of all commutators of elements of the free group on S
|
|-
| the symmetric group, S_n
| generators: $\backslash sigma\_1,\; \backslash ldots,\; \backslash sigma\_\{n-1\}$
relations:

• $\{\backslash sigma\_i\}^2\; =\; 1$,

• $\backslash sigma\_i\backslash sigma\_j\; =\; \backslash sigma\_j\backslash sigma\_i\; \backslash mbox\{\; if\; \}\; j\; \backslash neq\; i\backslash pm\; 1$,

• $\backslash sigma\_i\backslash sigma\_\{i+1\}\backslash sigma\_i\; =\; \backslash sigma\_\{i+1\}\backslash sigma\_i\backslash sigma\_\{i+1\}\backslash$

The last set of relations can be transformed into

• $\{(\backslash sigma\_i\backslash sigma\_\{i+1\}\})^3=1\backslash$

using $\{\backslash sigma\_i\}^2=1$.
| Here $\backslash sigma\_i$ is the permutation that swaps the ith element with the i+1 one. The product $\backslash sigma\_i\; \backslash sigma\_\{i+1\}\backslash$ is a 3-cycle on the set $\backslash \{i,i+1,i+2\backslash \}$.
|-
| the braid group, B_n
| generators: $\backslash sigma\_1,\; \backslash ldots,\; \backslash sigma\_\{n-1\}$
relations:

• $\backslash sigma\_i\backslash sigma\_j\; =\; \backslash sigma\_j\backslash sigma\_i\; \backslash mbox\{\; if\; \}\; j\; \backslash neq\; i\backslash pm\; 1$,

• $\backslash sigma\_i\backslash sigma\_\{i+1\}\backslash sigma\_i\; =\; \backslash sigma\_\{i+1\}\backslash sigma\_i\backslash sigma\_\{i+1\}\backslash$

| Note the similarity with the symmetric group; the only difference is the removal of the relation $\{\backslash sigma\_i\}^2\; =\; 1$.
|-
| the tetrahedral group, T ≅ A₄
| $\backslash langle\; s,t\; \backslash mid\; s^2,\; t^3,\; (st)^3\; \backslash rangle\backslash ,\backslash !$
|
|-
| the octahedral group, O ≅ S₄
| $\backslash langle\; s,t\; \backslash mid\; s^2,\; t^3,\; (st)^4\; \backslash rangle\backslash ,\backslash !$
|
|-
| the icosahedral group, I ≅ A₅
| $\backslash langle\; s,t\; \backslash mid\; s^2,\; t^3,\; (st)^5\; \backslash rangle\backslash ,\backslash !$
|
|-
| the Quaternion group, Q
| $\backslash langle\; i,j\; \backslash mid\; i^4,\; i^2j^2,\; ijij^\{-1\}\; \backslash rangle\backslash ,\backslash !$
|
|-
|$SL\_2(\backslash mathbb\{Z\})$
|$\backslash langle\; a,b\; \backslash mid\; aba=bab,\; (aba)^4\; \backslash rangle\backslash ,\backslash !$
|topologically you can visualize a and b as Dehn twists on the torus
|-
|$GL\_2(\backslash mathbb\{Z\})$
|$\backslash langle\; a,b,j\; \backslash mid\; aba=bab,\; (aba)^4,j^2,(ja)^2,(jb)^2\; \backslash rangle\backslash ,\backslash !$
|
|-
|$PSL\_2(\backslash mathbb\{Z\})$
|$\backslash langle\; a,b\; \backslash mid\; a^2,\; b^3\; \backslash rangle\backslash ,\backslash !$
|PSL₂(Z) is the free product of the cyclic groups Z₂ and Z₃
|-
|Heisenberg group
|$\backslash langle\; x,y,z\; \backslash mid\; z=xyx^\{-1\}y^\{-1\},\; xz=zx,\; yz=zy\; \backslash rangle\backslash ,\backslash !$
|
|-
|Baumslag-Solitar group, B(m,n)
|$\backslash langle\; a,\; b\; \backslash mid\; a^n\; =\; b\; a^m\; b^\{-1\}\; \backslash rangle\backslash ,\backslash !$
|
|-
| Tits group
| $a^2\; =\; b^3\; =\; (ab)^\{13\}\; =\; [a,\; b]^5\; =\; [a,\; bab]^4\; =\; (ababababab^\{-1\})^6\; =\; 1,$
|
|}

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.

Constructions

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>.

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.

Geometric group theory

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. These are also two resulting orders, the weak order and the Bruhat order, and corresponding Hasse diagrams. An important example is in the Coxeter groups.

Further, some properties of this graph (the coarse geometry) are intrinsic, meaning independent of choice of generators.

See also

• Nielsen transformation

• Tietze transformation

• Schur multiplier