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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a presentation is one method of specifying a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
. A presentation of a group ''G'' comprises a set ''S'' of generatorsŌĆöso that every element of the group can be written as a product of powers of some of these generatorsŌĆöand 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
to the
quotient In arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ß╝ĆŽü╬╣╬Ė╬╝ŽīŽé#Ancient Greek, ß╝ĆŽü╬╣╬Ė╬╝ŽīŽé ''arithmos'', 'number' and wikt:en:Žä╬╣╬║╬«#Ancient Greek, Žä╬╣╬║╬« wikt:en:Žä╬ŁŽć╬Į╬Ę#Ancient Greek, ä╬ŁŽć╬Į╬Ę ''tik├® ├®chne' ...
of a
free group for the free group on two generators would look like. Each vertex represents an element of the free group, and each edge represents multiplication by ''a'' or ''b''. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the stud ...
on ''S'' by the normal subgroup generated by the relations ''R''. As a simple example, the
cyclic group In group theory The popular puzzle Rubik's cube invented in 1974 by Ern┼æ Rubik has been used as an illustration of permutation group">Ern┼æ_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ern┼æ Rubik">Rubik's cube invented in 1974 by Er ... 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 :$\langle a \mid a^n\rangle,$ 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 groupIn mathematics, an absolute presentation is one method of defining a group (mathematics), group.B. Neumann, ''The isomorphism problem for algebraically closed groups,'' in: Word Problems, Decision Problems, and the Burnside Problem in Group Theory, A ...
.

# Background

A
free group for the free group on two generators would look like. Each vertex represents an element of the free group, and each edge represents multiplication by ''a'' or ''b''. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the stud ...
on a set ''S'' is a group where each element can be ''uniquely'' described as a finite length product of the form: :$s_1^ s_2^ \cdots s_n^$ where the ''si'' are elements of S, adjacent ''si'' are distinct, and ''ai'' 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 In mathematics, a dihedral group is the group (mathematics), group of symmetry, symmetries of a regular polygon, which includes rotational symmetry, rotations and reflection symmetry, reflections. Dihedral groups are among the simplest example ...
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. However, we have, for example, , , etc., so such products are ''not unique'' in D8. Each such product equivalence can be expressed as an equality to the identity, such as :, :, or :. Informally, we can consider these products on the left hand side as being elements of the free group , 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''ŌłÆ1''Rx'' of ''R'', then it follows by definition that every element of ''N'' is a finite product ''x''1ŌłÆ1''r''1''x''1 ... ''xm''ŌłÆ1''rm'' ''xm'' of members of such conjugates. 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 A quotient group or factor group is a mathematical group (mathematics), group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factore ...
. We then say that D8 has presentation :$\langle r, f \mid r^8 = 1, f^2 = 1, \left(rf\right)^2 = 1\rangle.$ Here the set of generators is , and the set of relations is . We often see ''R'' abbreviated, giving the presentation :$\langle r, f \mid r^8 = f^2 = \left(rf\right)^2 = 1\rangle.$ An even shorter form drops the equality and identity signs, to list just the set of relators, which is . Doing this gives the presentation :$\langle r, f \mid r^8, f^2, \left(rf\right)^2 \rangle.$ All three presentations are equivalent.

# Notation

Although the notation 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: * * * *

# Definition

Let ''S'' be a set and let ''FS'' be the
free group for the free group on two generators would look like. Each vertex represents an element of the free group, and each edge represents multiplication by ''a'' or ''b''. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the stud ...
on ''S''. Let ''R'' be a set of
words In linguistics Linguistics is the scientific study of language A language is a structured system of communication used by humans, including speech (spoken language), gestures (Signed language, sign language) and writing. Most lang ...
on ''S'', so ''R'' naturally gives a subset of $F_S$. To form a group with presentation $\langle S \mid R\rangle$, take the quotient of $F_S$ by the smallest normal subgroup that contains each element of ''R''. (This subgroup is called the normal closure ''N'' of ''R'' in $F_S$.) The group $\langle S \mid R\rangle$ is then defined as the
quotient group A quotient group or factor group is a mathematical group (mathematics), group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factore ...
:$\langle S \mid R \rangle = F_S / N.$ The elements of ''S'' are called the generators of $\langle S \mid R\rangle$ and the elements of ''R'' are called the relators. A group ''G'' is said to have the presentation $\langle S \mid R\rangle$ if ''G'' is isomorphic to $\langle S \mid R\rangle$. 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 $y^x\in R$. This has the intuitive meaning that the images of ''x'' and ''y'' are supposed to be equal in the quotient group. Thus, for example, ''rn'' in the list of relators is equivalent with $r^n=1$. For a finite group ''G'', it is possible to build a presentation of ''G'' from the group multiplication table, as follows. Take ''S'' to be the set elements $g_i$ of ''G'' and ''R'' to be all words of the form $g_ig_jg_k^$, where $g_ig_j=g_k$ is an entry in the multiplication table.

## Alternate definition

The definition of group presentation may alternatively be recast in terms of
equivalence class In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
es of words on the alphabet $S \cup S^$. 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 $x x^$ or $x^ x$ for some in , or by adding or removing a consecutive copy of a relator. The group elements are the equivalence classes, and the group operation is concatenation. This point of view is particularly common in the field of
combinatorial group theoryIn mathematics, combinatorial group theory is the theory of free groups, and the concept of a presentation of a group by generator (mathematics), generators and relation (mathematics), relations. It is much used in geometric topology, the fundamental ...
.

# Finitely presented groups

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, ) 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
G├Čdel numbering In mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical ...
) 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 ''FS''
recursive Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics Linguistics is the science, scientific study of language. It e ...
(respectively
recursively enumerable In computability theory, traditionally called recursion theory, a set ''S'' of natural numbers is called recursively enumerable, computably enumerable, semidecidable, partially decidable, listable, provable or Turing-recognizable if: *There is a ...
) 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 Graham Higman Fellow of the Royal Society, FRS (19 January 1917 – 8 April 2008) was a prominent English mathematician known for his contributions to group theory. Biography Higman was born in Louth, Lincolnshire and attended Sutton High Sc ... 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 In mathematics, a countable set is a Set (mathematics), set with the same cardinality (cardinal number, number of elements) as some subset of the set of natural numbers. A countable set is either a finite set or a countably infinite set. Whether f ...
many finitely generated recursively presented groups.
Bernhard Neumann Bernhard Hermann Neumann Companion of the Order of Australia, AC Fellow of the Royal Society, FRS (15 October 1909 ŌĆō 21 October 2002) was a Germany, German-born United Kingdom, British-Australian mathematician who was a leader in the study of grou ... has shown that there are
uncountably In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many Element (mathematics), elements to be countable set, countable. The uncountability of a set is closely related to its cardinal number: a set ...
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 Sir William Rowan Hamilton LL.D, DCL, MRIA (4 August 1805 ŌĆō 2 September 1865) was an Irish mathematician, Andrews Professor of Astronomy at Trinity College Dublin, Trinity College Dublin, and Dunsink Observatory#Directors, Royal Astronomer ...
in 1856, in his icosian calculus ŌĆō a presentation of the
icosahedral group A regular icosahedron has 60 rotational (or orientation-preserving) symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation. A regular dodecahedron has the same set of symmetries, since it is th ...
. The first systematic study was given by
Walther von Dyck Walther Franz Anton von Dyck (6 December 1856 ŌĆō 5 November 1934), born Dyck and later von, ennobled, was a Germany, German mathematician. He is credited with being the first to define a mathematical group (mathematics), group, in the modern sen ...
, student of
Felix Klein Christian Felix Klein (; 25 April 1849 ŌĆō 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group ...
, in the early 1880s, laying the foundations for
combinatorial group theoryIn mathematics, combinatorial group theory is the theory of free groups, and the concept of a presentation of a group by generator (mathematics), generators and relation (mathematics), relations. It is much used in geometric topology, the fundamental ...
.

# 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. An example of a
finitely generated group In algebra Algebra (from ar, ž¦┘äž¼ž©ž▒, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
that is not finitely presented is the
wreath product In group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( ...
$\mathbf \wr \mathbf$ of the group of
integers An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of t ... with itself.

# Some theorems

Theorem. Every group has a presentation.
To see this, given a group ''G'', consider the free group ''FG'' on ''G''. By the
universal property In category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...
of free groups, there exists a unique
group homomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ... whose restriction to ''G'' is the identity map. Let ''K'' be the
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...
of this homomorphism. Then ''K'' is normal in ''FG'', therefore is equal to its normal closure, so . Since the identity map is surjective, ''Žå'' is also surjective, so by the
First Isomorphism Theorem In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for ...
, . This presentation may be highly inefficient if both ''G'' and ''K'' are much larger than necessary.
Corollary. Every finite group has a finite presentation.
One may take the elements of the group for generators and the
Cayley table Named after the 19th century British British may refer to: Peoples, culture, and language * British people, nationals or natives of the United Kingdom, British Overseas Territories, and Crown Dependencies. ** Britishness, the British identity a ...
for relations.

## NovikovŌĆōBoone theorem

The negative solution to the
word problem for groups In mathematics, especially in the area of abstract algebra known as combinatorial group theory, the word problem for a finitely generated group ''G'' is the algorithmic problem of deciding whether two words in the generators represent the same elem ...
states that there is a finite presentation 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 200px, ''P. S. Novikov''. Pyotr Sergeyevich Novikov (russian: ą¤čæčéčĆ ąĪąĄčĆą│ąĄ╠üąĄą▓ąĖčć ąØąŠ╠üą▓ąĖą║ąŠą▓; 15 August 1901, Moscow, Russian Empire ŌĆō 9 January 1975, Moscow, Soviet Union) was a USSR, Soviet mathematician. Novikov is known for hi ...
in 1955 and a different proof was obtained by William Boone in 1958.

# Constructions

Suppose ''G'' has presentation and ''H'' has presentation with ''S'' and ''T'' being disjoint. Then * the
free product In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
has presentation and * the
direct productIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
has presentation , where 'S'', ''T''means that every element from ''S'' commutes with every element from ''T'' (cf.
commutator In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
).

# Deficiency

The deficiency of a finite presentation is just 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 ŌłÆdef(''G'') generators, and ''G'' is efficient if this number is required.

# Geometric group theory

A presentation of a group determines a geometry, in the sense of
geometric group theory Geometric group theory is an area in mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mat ...
: one has the
Cayley graph on two generators ''a'' and ''b'' In mathematics, a Cayley graph, also known as a Cayley colour graph, Cayley diagram, group diagram, or colour group is a Graph (discrete mathematics), graph that encodes the abstract structure of a group (mathemati ... , which has a
metric METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of or th ...
, called the
word metricIn group theory The popular puzzle Rubik's cube invented in 1974 by Ern┼æ Rubik has been used as an illustration of permutation group">Ern┼æ_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ern┼æ Rubik">Rubik's cube invented in 1974 by Ern┼ ...
. These are also two resulting orders, the ''weak order'' and the ''
Bruhat orderIn mathematics, the Bruhat order (also called strong order or strong Bruhat order or Chevalley order or BruhatŌĆōChevalley order or ChevalleyŌĆōBruhat order) is a partial order on the elements of a Coxeter group, that corresponds to the inclusion o ...
'', and corresponding
Hasse diagram In order theory Order theory is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they ...
s. An important example is in the
Coxeter group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s. Further, some properties of this graph (the coarse geometry) are intrinsic, meaning independent of choice of generators.

* Nielsen transformation * Tietze transformation * Presentation of a module * Presentation of a monoid

# References

* ŌĆĢ This useful reference has tables of presentations of all small finite groups, the reflection groups, and so forth. * ŌĆĢ Schreier's method, Nielsen's method, free presentations, subgroups and HNN extensions, GolodŌĆōShafarevich theorem, etc. * ŌĆĢ fundamental algorithms from theoretical computer science, computational number theory, and computational commutative algebra, etc.