Presentation of a group
{{#invoke:Hatnote|hatnote}} 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
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
where 1 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.
Contents
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 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}Rx of R, then it is straightforward to show that every element of N is a finite product x_{1}^{−1}r_{1}x_{1} ... 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
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 to take F_{S} quotient by the smallest normal subgroup such that each element of R gets identified with the identity. Note that R might not be a subgroup, let alone a normal subgroup of F_{S}, so we cannot take a quotient by R. The solution is to take the normal closure N of R in F_{S}. 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 ∈ 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 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} → 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 , where 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 | A free group is "free" in the sense that it is subject to no relations. | |
C_{n}, the cyclic group of order n | ||
D_{2n}, the dihedral group of order 2n | Here r represents a rotation and f a reflection | |
D_{∞}, the infinite dihedral group | ||
Dic_{n}, the dicyclic group | The quaternion group is a special case when n = 2 | |
Z × Z | ||
Z/mZ × Z/nZ | ||
the free abelian group on S | where R is the set of all commutators of elements of S | |
the symmetric group, S_{n} | generators: relations: The last set of relations can be transformed into |
Here σ_{i} is the permutation that swaps the ith element with the i+1 one. The product σ_{i}σ_{i+1} is a 3-cycle on the set {i, i+1, i+2}. |
the braid group, B_{n} | generators: relations: |
Note the similarity with the symmetric group; the only difference is the removal of the relation . |
the tetrahedral group, T ≅ A_{4} | ||
the octahedral group, O ≅ S_{4} | ||
the icosahedral group, I ≅ A_{5} | ||
the quaternion group, Q_{8} | For an alternative presentation see Dic_{n} above. | |
SL(2, Z) | topologically you can visualize a and b as Dehn twists on the torus | |
GL(2, Z) | nontrivial Z/2Z – group extension of SL(2, Z) | |
the modular group PSL(2, Z) | PSL(2, Z) is the free product of the cyclic groups Z/2Z and Z/3Z | |
Heisenberg group | ||
Baumslag–Solitar group, B(m, n) | ||
Tits group | [a, b] is the commutator |
An example of a finitely generated group that is not finitely presented is the wreath product of the group of integers with itself.
Some theorems
Theorem. Every group has a presentation.
To see this, given a group G, 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> ≅ im(φ) = G. Note that 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 for relations.
Novikov–Boone theorem
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. This was shown by Pyotr Novikov in 1955^{[3]} and a different proof was obtained by William Boone in 1958.^{[4]}
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).
Deficiency
The deficiency of a finite presentation <S|R> is just |S|−|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 group G can be generated by −def G generators, and G is efficient if this number is required.^{[5]}
Geometric group theory
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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
Notes
References
- {{#invoke:citation/CS1|citation
|CitationClass=book }} Schreier's method, Nielsen's method, free presentations, subgroups and HNN extensions, Golod–Shafarevich theorem, etc.
- {{#invoke:citation/CS1|citation
|CitationClass=book }} This useful reference has tables of presentations of all small finite groups, the reflection groups, and so forth.