Presentation of a group

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 powers 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

$\langle a \mid a^n = 1\rangle.\,\!$

where 1 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.

1 Background
2 Formal definition
3 Examples
- 3.1 History
- 3.2 Common examples
4 Some theorems
5 Constructions
6 Geometric group theory
7 See also
8 Notes
9 References

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 the s_i are elements of S, adjacent s_i are distinct, and a_i are non-zero integers (but n may be zero). 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 r 's and f 's.

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

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 D.

If we then let N be the subgroup of F generated by all conjugates x ⁻¹ R x of R, then it is straightforward to show that every element of N is a finite product x₁ ⁻¹ r₁ x₁ . . . x_m ⁻¹ r_m x_m of members of such conjugates. It follows that N is a normal subgroup of F; and that 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

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

Formal definition

Let S be a set and let F_S be the free group on S. Let R be a set of words on S, so R naturally gives a subset of F_S. To form a group with presentation <S|R> the idea is take the smallest quotient of F_S (that is, the largest subgroup of F_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 F_S, so we cannot take a naive quotient by R. The solution is to take the normal closure N of R in F_S, which is defined as the smallest normal subgroup in F_S which contains R. The group <S|R> is then defined as the quotient group

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

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. $r^n$ 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 $\mathbb{N}$ or a finite subset of them, then it is easy to set up a simple one to one coding (or Gödel numbering) $f:F_S\rightarrow\mathbb{N}$ 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 $F_S$ 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 $g_i$ 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 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.^[1]

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.^[2]

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.

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.
C_n, the cyclic group of order n	$\langle a \mid a^n \rangle\,\!$
D_2n, 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	$\langle r, f \mid r^2, f^2 \rangle\,\!$
Dic_n, 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	$\langle x, y \mid xy=yx \rangle\,\!$
Z_m × Z_n	$\langle x, y \mid x^m=1, y^n=1, xy=yx \rangle\,\!$
the free abelian group on S	$\langle S \mid R \rangle\,\!$ where R is the set of all commutators of elements of S
the symmetric group, S_n	generators: $\sigma_1, \ldots, \sigma_{n-1}$ relations: ${\sigma_i}^2 = 1$ , $\sigma_i\sigma_j = \sigma_j\sigma_i \mbox{ if } j \neq i\pm 1$ , $\sigma_i\sigma_{i%2B1}\sigma_i = \sigma_{i%2B1}\sigma_i\sigma_{i%2B1}\$ The last set of relations can be transformed into ${(\sigma_i\sigma_{i%2B1}})^3=1\$ using ${\sigma_i}^2=1$ .	Here $\sigma_i$ is the permutation that swaps the ith element with the i+1 one. The product $\sigma_i \sigma_{i%2B1}\$ is a 3-cycle on the set $\{i,i%2B1,i%2B2\}$ .
the braid group, B_n	generators: $\sigma_1, \ldots, \sigma_{n-1}$ relations: $\sigma_i\sigma_j = \sigma_j\sigma_i \mbox{ if } j \neq i\pm 1$ , $\sigma_i\sigma_{i%2B1}\sigma_i = \sigma_{i%2B1}\sigma_i\sigma_{i%2B1}\$	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, Q₈	$\langle i,j \mid jij = i, iji = j \rangle\,$	For an alternative presentation see Dic_n above.
$SL_2(\mathbb{Z})$	$\langle a,b \mid aba=bab, (aba)^4 \rangle\,\!$	topologically you can visualize a and b as Dehn twists on the torus
$GL_2(\mathbb{Z})$	$\langle a,b,j \mid aba=bab, (aba)^4,j^2,(ja)^2,(jb)^2 \rangle\,\!$	non trivial $\mathbb{Z}_2$ - group extension of $SL_2(\mathbb{Z})$
$PSL_2(\mathbb{Z})$	$\langle a,b \mid a^2, b^3 \rangle\,\!$	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)	$\langle a, b \mid a^n = b a^m b^{-1} \rangle\,\!$
Tits group	$\langle a, b \mid a^2, b^3, (ab)^{13}, [a, b]^5, [a, bab]^4, (ababababab^{-1})^6 \rangle$	[a, b] is the commutator

An example of a finitely generated group that is not finitely presented is the wreath product $\mathbb{Z} \wr \mathbb{Z}$ of the group of integers with itself.

Some theorems

Every group G has a presentation. To see this, consider the free group F_G on G. By the universal property of free groups, there exists a unique group homomorphism φ : F_G → G whose restriction to G is the identity map. Let K be the kernel of this homomorphism. Then K is normal in F_G, therefore is equal to its normal closure, so <G|K> = F_G/K. Since the identity map is surjective, φ is also surjective, so by the First Isomorphism Theorem, <G|K> $\cong \operatorname{im} \varphi$ = G. Note that this presentation may be highly inefficient if 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

Suppose 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> and
the direct product G × H has presentation <S,T|R,Q, [S,T]>, where [S,T] means that every element from S commutes with every element from T (cf. commutator).

Geometric group theory

Main article: Geometric group theory

Further information: Cayley graph

Further information: Word metric

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.

Notes

^ Sir William Rowan Hamilton (1856). "Memorandum respecting a new System of Roots of Unity". Philosophical Magazine 12: 446. http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Icosian/NewSys.pdf.
^ Stillwell, John (2002). Mathematics and its history. Springer. p. 374. ISBN 978-0-38795336-6

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.