Editing Presentation of a group

{{Short description|Specification of a mathematical group by generators and relations}}
{{About|relations of a mathematical group|describing a module over a ring|Free presentation|topics in mathematical representation theory|Group representation}}
{{Redirect|Relator|the Catholic position preparing the cases of candidates for sainthood |Dicastery for the Causes of Saints#Congregation for the Causes of Saints}}
In [[mathematics]], a '''presentation''' is one method of specifying a [[group (mathematics)|group]]. A presentation of a group ''G'' comprises a '''[[generating set of a group|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

:<math>\langle S \mid R\rangle.</math>

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 [[group isomorphism|isomorphic]] to the [[quotient group|quotient]] of a [[free group]] on ''S'' by the [[Normal closure (group theory)|normal subgroup generated by]] the relations ''R''.

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

:<math>\langle a \mid a^n = 1\rangle,</math>

where 1 is the group identity.  This may be written equivalently as

:<math>\langle a \mid a^n\rangle,</math>

thanks to the convention that terms that do not include an equals sign are taken to be equal to the group identity.  Such terms are called '''relators''', distinguishing them from the relations that do 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:

:<math>s_1^{a_1} s_2^{a_2} \cdots s_n^{a_n}</math>

where the ''s<sub>i</sub>'' are elements of S, adjacent ''s<sub>i</sub>'' are distinct, and ''a<sub>i</sub>'' 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 an adjacent occurrence of 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<sub>8</sub> 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<sub>8</sub> is a product of ''r''{{'}}s and ''f''{{'}}s.

However, we have, for example, {{math|1=''rfr'' = ''f''<sup>−1</sup>}}, {{math|1=''r''<sup>7</sup> = ''r''<sup>−1</sup>}}, etc., so such products are ''not unique'' in D<sub>8</sub>. Each such product equivalence can be expressed as an equality to the identity, such as

:{{math|1=''rfrf'' = 1}}, 
:{{math|1=''r''<sup>8</sup> = 1}}, or
:{{math|1=''f''{{px2}}<sup>2</sup> = 1}}.

Informally, we can consider these products on the left hand side as being elements of the free group {{math|1=''F'' = ⟨''r'', ''f''&nbsp;⟩}}, and let {{math|1=''R'' = ⟨''rfrf'', ''r''<sup>8</sup>, ''f''{{px2}}<sup>2</sup>⟩}}. That is, we let ''R'' be the subgroup generated by the strings ''rfrf'', ''r''<sup>8</sup>, ''f''{{px2}}<sup>2</sup>, each of which is also equivalent to 1 when considered as products in D<sub>8</sub>.

If we then let ''N'' be the subgroup of ''F'' generated by all conjugates ''x''<sup>−1</sup>''Rx'' of ''R'', then it follows by definition that every element of ''N'' is a finite product ''x''<sub>1</sub><sup>−1</sup>''r''<sub>1</sub>''x''<sub>1</sub> ... ''x<sub>m</sub>''<sup>−1</sup>''r<sub>m</sub>'' ''x<sub>m</sub>'' of members of such conjugates. It follows that each element of ''N'', when considered as a product in D<sub>8</sub>, will also evaluate to 1; and thus that ''N'' is a normal subgroup of ''F''. Thus D<sub>8</sub> is isomorphic to the [[quotient group]] {{math|1=''F''/''N''}}. We then say that D<sub>8</sub> has presentation

:<math>\langle r, f \mid r^8 = 1, f^2 = 1, (rf)^2 = 1\rangle.</math>

Here the set of generators is {{math|1=''S'' = {{mset|''r'', ''f''&nbsp;}}}}, and the set of relations is {{math|1=''R'' = {''r'' <sup>8</sup> = 1, ''f'' <sup>2</sup> = 1, (''rf'' )<sup>2</sup> = 1} }}.  We often see ''R'' abbreviated, giving the presentation

:<math>\langle r, f \mid r^8 = f^2 = (rf)^2 = 1\rangle.</math>

An even shorter form drops the equality and identity signs, to list just the set of relators, which is {{math|1= {''r'' <sup>8</sup>, ''f'' <sup>2</sup>, (''rf'' )<sup>2</sup>} }}.  Doing this gives the presentation

:<math>\langle r, f \mid r^8, f^2, (rf)^2 \rangle.</math>

All three presentations are equivalent.

== Notation ==
Although the notation {{math|{{braket|bra-ket|''S'' | ''R''}}}}  used in this article for a presentation is now the most common, earlier writers used different variations on the same format.  Such notations include the following:{{cn|date=August 2020}}
*{{math|{{braket|bra-ket|''S'' | ''R''}}}}
*{{math|(''S'' {{!}} ''R'')}}
*{{math|{{mset|''S''; ''R''}}}}
*{{math|{{angbr|''S''; ''R''}}}}

== Definition ==
{{redirect|Relator}}
Let ''S'' be a set and let ''F<sub>S</sub>'' be the [[free group]] on ''S''.  Let ''R'' be a set of [[Word (group theory)|words]] on ''S'', so ''R'' naturally gives a subset of <math>F_S</math>. To form a group with presentation <math>\langle S \mid R\rangle</math>, take the quotient of <math>F_S</math> by the smallest normal subgroup that contains each element of ''R''.  (This subgroup is called the [[Normal closure (group theory)|normal closure]] ''N'' of ''R'' in <math>F_S</math>.)  The group <math>\langle S \mid R\rangle</math> is then defined as the [[quotient group]]
:<math>\langle S \mid R \rangle = F_S / N.</math>
The elements of ''S'' are called the '''generators''' of <math>\langle S \mid R\rangle</math> and the elements of ''R'' are called the '''relators'''. A group ''G'' is said to have the presentation <math>\langle S \mid R\rangle</math> if ''G'' is isomorphic to <math>\langle S \mid R\rangle</math>.<ref name = Peifer>{{cite journal|first = David|last = Peifer|title = An Introduction to Combinatorial Group Theory and the Word Problem|journal = Mathematics Magazine | volume = 70 | issue = 1 | pages = 3–10 | doi = 10.1080/0025570X.1997.11996491 | year = 1997}}</ref>

It is a common practice to write relators in the form <math>x = y</math> where ''x'' and ''y'' are words on ''S''. What this means is that <math>y^{-1}x\in R</math>. This has the intuitive meaning that the images of ''x'' and ''y'' are supposed to be equal in the quotient group. Thus, for example, ''r<sup>n</sup>'' in the list of relators is equivalent with <math>r^n=1</math>.<ref name = Peifer />

For a finite group ''G'', it is possible to build a presentation of ''G'' from the [[Cayley table|group multiplication table]], as follows. Take ''S'' to be the set elements <math>g_i</math> of ''G'' and ''R'' to be all words of the form <math>g_ig_jg_k^{-1}</math>, where <math>g_ig_j=g_k</math> is an entry in the multiplication table.

=== Alternate definition ===
The definition of group presentation may alternatively be recast in terms of [[equivalence class]]es of words on the alphabet <math>S \cup S^{-1}</math>.  In this perspective, we declare two words to be equivalent if it is possible to get from one to the other by a sequence of moves, where each move consists of adding or removing a consecutive pair <math>x x^{-1}</math> or <math>x^{-1} x</math> for some {{mvar|x}} in {{mvar|S}}, or by adding or removing a consecutive copy of a relator.  The group elements are the equivalence classes, and the group operation is concatenation.<ref name = Peifer />

This point of view is particularly common in the field of [[combinatorial group theory]].

== Finitely presented groups ==
A presentation is said to be '''[[finitely generated group|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''', '''{{visible anchor|finitely presented}}''') if it has a presentation that is finitely generated (respectively finitely related, a finite presentation). A group which has a finite presentation with a single relation is called a '''one-relator group'''.

== Recursively presented groups ==
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]]) {{nowrap|''f'' : ''F<sub>S</sub>'' → '''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<sub>S</sub>'' [[recursive set|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.

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.<ref>{{Cite journal |date=1961-08-08 |title=Subgroups of finitely presented groups |url=https://royalsocietypublishing.org/doi/10.1098/rspa.1961.0132 |journal=Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences |language=en |volume=262 |issue=1311 |pages=455–475 |doi=10.1098/rspa.1961.0132 |bibcode=1961RSPSA.262..455H |s2cid=120100270 |issn=0080-4630 |last1=Higman |first1=G. }}</ref> 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.

== 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]].<ref>{{Cite journal |title=Memorandum respecting a new System of Roots of Unity |author=Sir William Rowan Hamilton |author-link=William Rowan Hamilton |url=http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Icosian/NewSys.pdf |archive-url=https://web.archive.org/web/20030626113045/http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Icosian/NewSys.pdf |archive-date=2003-06-26 |url-status=live |journal=[[Philosophical Magazine]] |volume=12 |year=1856 |pages=446 }}</ref>
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]].<ref name="stillwell374">{{Cite book | publisher = Springer | isbn = 978-0-387-95336-6 | last = Stillwell | first = John | title = Mathematics and its history | date = 2002 | page = [https://books.google.com/books?id=WNjRrqTm62QC&pg=PA374 374] }}</ref>

== 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 class="wikitable"
!Group || Presentation || Comments
|-
| the [[free group]] on ''S'' || <math>\langle S \mid \varnothing \rangle</math>
| A free group is "free" in the sense that it is subject to no relations.
|-
|<math>\pi_1(S_g)</math>, the [[surface group]] of orientable genus <math>g\ge 0</math>
|<math>\left\langle a_1, b_1,\ldots, a_g, b_g | [a_1, b_1][a_2, b_2]\ldots [a_g, b_g] \right\rangle</math>
|The bracket stands for the commutator: <math>[a, b] = aba^{-1}b^{-1}</math>
|-
| C<sub>''n''</sub>, the [[cyclic group]] of order ''n''
| <math>\langle a \mid a^n \rangle</math>
|
|-
| D<sub>''n''</sub>, the [[dihedral group]] of order 2''n''
| <math>\langle r,f \mid r^n , f^2 , (rf)^2 \rangle</math>
| Here ''r'' represents a rotation and ''f'' a reflection 
|-
| D<sub>∞</sub>, the [[infinite dihedral group]]
| <math>\langle r,f \mid f^2, (rf)^2 \rangle</math>
|
|-
| Dic<sub>''n''</sub>, the [[dicyclic group]]
| <math>\langle r,f \mid r^{2n}, r^n=f^2, frf^{-1}=r^{-1} \rangle</math>
| The [[quaternion group]] Q<sub>8</sub> is a special case when ''n'' = 2
|-
| '''Z''' × '''Z'''
| <math>\langle x,y \mid xy = yx \rangle</math>
|
|-
| '''Z'''/''m'''''Z''' × '''Z'''/''n'''''Z'''
| <math>\langle x,y \mid x^m, y^n, xy=yx \rangle</math>
|
|-
| the [[free abelian group]] on ''S''
| <math>\langle S \mid R \rangle</math> where ''R'' is the set of all [[commutator]]s of elements of ''S''
|
|-
| S<sub>''n''</sub>, the [[symmetric group]] on ''n'' symbols
| generators: <math>\sigma_1, \ldots, \sigma_{n-1}</math><br>relations:
*<math>\sigma_i^2 = 1</math>,
*<math>\sigma_i\sigma_j = \sigma_j\sigma_i \mbox{ if } j \neq i\pm 1</math>,
*<math>\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}\ </math>

The last set of relations can be transformed into 
*<math>{(\sigma_i\sigma_{i+1}})^3=1\ </math>
using <math>\sigma_i^2=1 </math>.
| Here ''σ''<sub>''i''</sub> is the permutation that swaps the ''i''th element with the ''i''+1st one. The product ''σ''<sub>''i''</sub>''σ''<sub>''i''+1</sub> is a 3-cycle on the set {''i'', ''i''+1, ''i''+2}.
|-
| B<sub>''n''</sub>, the [[braid group]]s
| generators: <math>\sigma_1, \ldots, \sigma_{n-1}</math><br>
relations:
*<math>\sigma_i\sigma_j = \sigma_j\sigma_i \mbox{ if } j \neq i\pm 1</math>,
*<math>\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}\ </math>
| Note the similarity with the symmetric group; the only difference is the removal of the relation <math>\sigma_i^2 = 1</math>.
|-
| {{nowrap|V<sub>4</sub> ≅ D<sub>2</sub>}}, the [[Klein 4 group]]
| <math>\langle s,t \mid s^2, t^2, (st)^2 \rangle</math>
|
|-
| {{nowrap|T ≅ A<sub>4</sub>}}, the [[tetrahedral group]]
| <math>\langle s,t \mid s^2, t^3, (st)^3 \rangle</math>
|
|-
| {{nowrap|O ≅ S<sub>4</sub>}}, the [[octahedral group]]
| <math>\langle s,t \mid s^2, t^3, (st)^4 \rangle</math>
|
|-
| {{nowrap|I ≅ A<sub>5</sub>}}, the [[Icosahedral symmetry|icosahedral group]]
| <math>\langle s,t \mid s^2, t^3, (st)^5 \rangle</math>
|
|-
| Q<sub>8</sub>, the [[quaternion group]]
| <math>\langle i,j \mid i^4, jij = i, iji = j \rangle\,</math>
| For an alternative presentation see Dic<sub>''n''</sub> above with n=2.
|-
| SL(2, '''Z''') 
| <math>\langle a,b \mid aba=bab, (aba)^4 \rangle</math>
| topologically ''a'' and ''b'' can be visualized as [[Dehn twist]]s on the [[torus]]
|-
| GL(2, '''Z''')
| <math>\langle a,b,j \mid aba=bab, (aba)^4,j^2,(ja)^2,(jb)^2 \rangle</math>
| nontrivial '''Z'''/2'''Z''' – [[group extension]] of SL(2, '''Z''') 
|-
| PSL(2, '''Z'''), the [[modular group]]
| <math>\langle a,b \mid a^2, b^3 \rangle</math>
| PSL(2, '''Z''') is the [[free product]] of the cyclic groups '''Z'''/2'''Z''' and '''Z'''/3'''Z'''
|-
| [[Heisenberg group]]
| <math>\langle x,y,z \mid z=xyx^{-1}y^{-1}, xz=zx, yz=zy \rangle</math>
|
|-
| BS(''m'', ''n''), the [[Baumslag–Solitar group]]s
| <math>\langle a, b \mid  a^n = b a^m b^{-1}  \rangle</math>
|
|-
| [[Tits group]]
| <math>\langle a, b \mid a^2, b^3, (ab)^{13}, [a, b]^5, [a, bab]^4, ((ab)^4 ab^{-1})^6 \rangle</math>
| [''a'', ''b''] is the [[commutator]]
|}

An example of a [[finitely generated group]] that is not finitely presented is the [[wreath product]] <math>\mathbf{Z} \wr \mathbf{Z}</math> of the group of [[integers]] with itself.

== Some theorems ==
<blockquote>'''Theorem.''' Every group has a presentation.</blockquote>

To see this, given a group ''G'', consider the free group ''F<sub>G</sub>'' on ''G''. By the [[universal property]] of free groups, there exists a unique [[group homomorphism]] {{math|φ : ''F<sub>G</sub>'' → ''G''}} whose restriction to ''G'' is the identity map. Let ''K'' be the [[kernel (algebra)|kernel]] of this homomorphism. Then ''K'' is normal in ''F<sub>G</sub>'', therefore is equal to its normal closure, so {{math|1=⟨''G'' {{!}} ''K''⟩ = ''F<sub>G</sub>''/''K''}}. Since the identity map is surjective, ''φ'' is also surjective, so by the [[Isomorphism theorems|First Isomorphism Theorem]], {{math|1=⟨''G'' {{!}} ''K''⟩ ≅ im(''φ'') = ''G''}}. This presentation may be highly inefficient if both ''G'' and ''K'' are much larger than necessary.

<blockquote>'''Corollary.''' Every finite group has a finite presentation.</blockquote>

One may take the elements of the group for generators and the [[Cayley table]] for relations.

===Novikov–Boone theorem===
The negative solution to the [[word problem for groups]] states that there is a finite presentation  {{math|⟨''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<ref>{{Citation|last=Novikov|first=Pyotr S.|author-link=Pyotr Novikov|year=1955|title=On the algorithmic unsolvability of the word problem in group theory|language=ru| zbl=0068.01301 | journal=[[Proceedings of the Steklov Institute of Mathematics]]|volume=44|pages=1–143}}</ref> and a different proof was obtained by [[William Boone (mathematician)|William Boone]] in 1958.<ref>{{Citation|last=Boone|first=William W.| author-link=William Boone (mathematician) | year=1958|title=The word problem|journal=[[Proceedings of the National Academy of Sciences]]|volume=44|issue=10|pages=1061–1065|url=http://www.pnas.org/cgi/reprint/44/10/1061.pdf |archive-url=https://web.archive.org/web/20150924133037/http://www.pnas.org/cgi/reprint/44/10/1061.pdf |archive-date=2015-09-24 |url-status=live|doi=10.1073/pnas.44.10.1061|zbl=0086.24701 |pmc=528693 |pmid=16590307|bibcode=1958PNAS...44.1061B|doi-access=free}}</ref>

== Constructions ==
Suppose ''G'' has presentation {{math|⟨''S'' {{!}} ''R''⟩}} and ''H'' has presentation {{math|⟨''T'' {{!}} ''Q''⟩}} with ''S'' and ''T'' being disjoint. Then
* the '''[[free product]]''' {{math|''G'' ∗ ''H''}} has presentation {{math|⟨''S'', ''T'' {{!}} ''R'', ''Q''⟩}};
* the '''[[direct product of groups|direct product]]''' {{math|''G'' × ''H''}} has presentation {{math|⟨''S'', ''T'' {{!}} ''R'', ''Q'', [''S'', ''T'']⟩}}, where [''S'', ''T''] means that every element from ''S'' commutes with every element from ''T'' (cf. [[commutator]]); and
* the '''[[semidirect product]]''' {{math|''G'' ⋊{{sub|''φ''}} ''H''}} has presentation {{math|⟨''S'', ''T'' {{!}} ''R'', ''Q'', {''t'' ''s'' ''t''<sup>−1</sup> ''φ''<sub>''t''</sub>(''s'')<sup>−1</sup> {{!}} ''s'' in ''S'', ''t'' in ''T''}⟩}}.<ref>{{cite book |last1=Johnson |first1=DL |title=Presentations of groups |date=1990 |publisher=Cambridge University Press |location=Cambridge, U.K. ; New York, NY, USA |isbn=9780521585422 |page=140 |url=https://archive.org/details/presentationsofg0000john_z8f6/page/140}}</ref>

==Deficiency==
The '''deficiency''' of a finite presentation {{math|⟨''S'' {{!}} ''R''⟩}} is just {{math|{{abs|''S''}} − {{abs|''R''}}}} and the ''deficiency'' of a finitely presented group ''G'', denoted def(''G''), is the maximum of the deficiency over all presentations of ''G''.  The deficiency of a finite group is non-positive.  The [[Schur multiplicator]] of a finite group ''G'' can be generated by −def(''G'') generators, and ''G'' is '''efficient''' if this number is required.<ref>{{cite book | first1=D.L. | last1=Johnson | first2=E.L. | last2=Robertson | chapter=Finite groups of deficiency zero | pages=275–289 | editor1-first=C.T.C. | editor1-last=Wall | editor-link=C. T. C. Wall | title=Homological Group Theory | series=London Mathematical Society Lecture Note Series | volume=36 | year=1979 | publisher=[[Cambridge University Press]] | isbn=0-521-22729-1 | zbl=0423.20029 }}</ref>

== 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 (mathematics)|metric]], called the [[word metric]]. These are also two resulting orders, the ''weak order'' and the ''[[Bruhat order]]'', and corresponding [[Hasse diagram]]s. An important example is in the [[Coxeter group]]s.

Further, some properties of this graph (the [[Coarse structure|coarse geometry]]) are intrinsic, meaning independent of choice of generators.

== See also ==
* [[Nielsen transformation]]
* [[Presentation of a module]]
* [[Presentation of a monoid]]
* [[Set-builder notation]]
* [[Tietze transformation]]

== Notes ==
{{Reflist}}

== References ==
*{{cite book | last1=Coxeter | first1=H. S. M. | author-link1=Harold Scott Macdonald Coxeter | last2=Moser | first2=W. O. J. | author-link2=William Oscar Jules Moser | title=Generators and Relations for Discrete Groups | location=New York | publisher=Springer-Verlag | year=1980 | isbn=0-387-09212-9}} ― This useful reference has tables of presentations of all small finite groups, the reflection groups, and so forth.
*{{cite book | last=Johnson | first=D. L. | author-link=D. L. Johnson | title=Presentations of Groups | location=Cambridge | publisher=Cambridge University Press | year=1997 | edition=2nd | isbn=0-521-58542-2}} ― Schreier's method, Nielsen's method, free presentations, subgroups and HNN extensions, [[Golod–Shafarevich theorem]], etc.
*{{cite book | last=Sims | first=Charles C. | author-link=Charles Sims (mathematician) | title=Computation with Finitely Presented Groups | url=https://archive.org/details/computationwithf0000sims | url-access=registration | location=Cambridge | publisher=Cambridge University Press | year=1994 | edition=1st | isbn=978-0-521-13507-8}} ― fundamental algorithms from theoretical computer science, computational number theory, and computational commutative algebra, etc.

== External links ==
*{{MathWorld|title=Group Presentation|id=GroupPresentation|author=[[Yves de Cornulier|de Cornulier, Yves]]}}
*[http://groupnames.org Small groups and their presentations on GroupNames]

[[Category:Combinatorial group theory]]
[[Category:Combinatorics on words]]