nLab group presentation

Presentations of groups

Presentations of groups


In trying to study a group GG, one way to proceed is to

  • look for a set of generating elements;

  • look for ‘relations’ between those elements.

The problem is partially how one is to interpret this second part. To do this we need to look at words in the generators and hence at the free group on the set of generators.

For example, if we have the symmetric group S 3S_3, or, isomorphically, the dihedral group (of symmetries of a triangle) which we will call D 3D_3 (following the geometric convention (see the Wikipedia article on dihedral groups), then we have 6 elements, and they are all able to be got as products of a transposition and a three cycle (or alternatively as a reflection and a rotation). If we call the three cycle aa and the transposition bb, we have

S 3={1,a,a 2,b,ab,a 2b}.S_3=\{1,a,a^2,b,a b,a^2b\}.

What about the other words: baba for instance. If one calculates bab a in S 3S_3 and then looks up what you get you have ba=a 2bba = a^2b, so that there is a relation between these two words. This however is not all. What about babaabaaabbaab a b a a b a a a b b a a ? That is a word in the aas and bbs so should represent something in S 3S_3. in fact if you think in the geometric interpretation of S 3S_3 as being the dihedral group, D 3D_3, you can pick up a triangle and interpret that word as a list of instructions for moving the triangle. You then find out what this word is out of the 6 possible given forms.

For a presentation, you give a set of generators XX, so there will be an epimorphism from F(X)F(X) to GG, and then you try to find a description for the kernel of that epimorphism, which we will denote by NN. The description of this normal subgroup NN is as the normal closure of a set RR of relations, i.e. words in the generators or, equally validly, elements in the free group on XX.


A presentation of a group, GG, is a coequalizer diagram FRFXGF R \rightrightarrows F X \twoheadrightarrow G, where FXF X is the free group on a set of generators, and FRF R one on a set of relations (or relators, depending on how the relations are specified).

This is not quite the usual `classical' form of the definition, so we will take it apart to show the relationship.

A presentation is thus given by a pair of sets, XX and RR, written X:R\langle X: R\rangle such that setting F=F(X)=X=X:F=F(X)= \langle X\rangle =\langle X:\emptyset\rangle to be the free group on the set XX and NN the normal closure of the set of relators RR, there is a specified isomorphism from F/NF/N to GG.

The specified isomorphism is often omitted, as often the set XX of generators is chosen as a subset of the set of elements of GG. In this case, the universal properties of free groups and quotients, there is a unique map FGF\to G which restricts to the inclusion of XX, and thereby at most one map F/NGF/N \to G which does so; this map is then the one asserted to be an isomorphism.

In general it is necessary to proceed otherwise, however, and to give a specific function from a set XX to the set of elements of GG. This function then induces a group homomorphism from F=XF=\langle X\rangle to GG, and if this is a surjection, then we can find some NN (generators for its kernel) to produce a presentation of GG. Without this extra data, certain of the manipulations of a group presntation look decidedly suspect, for instance a substitution which results in two copies of a generator being given. It is also much easier to work with morphisms of presentations if the full data is recorded.

Examples of presentations

The standard presentation

A standard if somewhat trivial example of a presentation is given by the standard presentation of a group, GG. We take X={x ggG,g1}X= \{x_g\mid g\in G,g\neq 1\}, to be a set in bijective correspondence with the underlying set of GG. (You can take XX equal to that set if you like, but sometimes it is better to have a distinct set, for instance, it make for an easier notation for the description of certain morphisms.) The set of relations will be

R={x g.x h=x ghg,hG},R =\{x_g.x_h= x_{gh}\mid g,h \in G\},

so as a set is just a copy of G×GG\times G as the relations are indexed by pairs of elements of GG.

Cyclic groups

Cyclic group, C nC_n, of order nn has presentation a:a n\langle a : a^n\rangle. There are many different functions from the (singleton) set of generators to C nC_n that will give a suitable presentation in the fuller sense.

The symmetric group of order 6

The group of permutations of three letters, S 3S_3, has a presentation

a,b:a 3,b 2,(ab) 2.\langle a,b : a^3, b^2,(a b)^2 \rangle.

Here aa corresponds to the 3-cycle, (123)(123), whilst bb corresponds to any transposition.

Knot groups

The trefoil knot group has two useful presentations:

  • a,b:a 3=b 2\langle a,b : a^3= b^2\rangle, which displays the fact that the trefoil is a (2,3)-torus knot; and

  • x,y:xyx=yxy\langle x,y : x y x = y x y\rangle, which shows the link between this group and the Artin braid group, Br3Br3.

Coxeter groups

At the entry on Coxeter groups one finds the following:

Definition A Coxeter matrix over an index set II is a symmetric matrix

M:I×I{1,2,3,,}M \colon I \times I \to \{1, 2, 3, \ldots, \infty\}

such that M(i,i)=1M(i, i) = 1 for all iIi \in I, else M(i,j)>1M(i, j) \gt 1. Writing m i,j=M(i,j)m_{i,j} = M(i, j), the associated Coxeter group W MW_M is the group presented as having generators s is_i, iIi \in I, and relations

(s is j) m i,j=1(s_i s_j)^{m_{i, j}} = 1

for all i,jIi, j \in I, whenever m i,jm_{i, j} \neq \infty. In other words, m i,jm_{i, j} is the order of s is js_i s_j (as is easily shown), and these orders determine the group.

We thus have an infinite family of group presentations.


  • ‘Relations’ and ‘relators’: In the discussion of S 3S_3 above we had a relation ba=a 2bb a = a^2b. so we are relating two words of F(X)F(X). It is often the case that instead of relations we use relators, in other words a relation of form r=1r = 1, where rr is a word in the generators. In the example ba=a 2b b a = a^2b can be easily shown to imply and be implied by abab=1 a b a b = 1.

  • Given a group presentation as above, we have a short exact sequence,

1NFG1,1\to N \to F \to G \to 1,

where F=F(X)F = F(X), the free group on the set XX, RR is a subset of FF and N=N(R)N = N(R) is the normal closure in FF of the set RR. The group FF acts on NN by conjugation: uc=ucu 1,forcN,uF{}^u c = ucu^{-1}, for c\in N, u \in F and the elements of NN are words in the conjugates of the elements of RR:

c= u 1(r 1 ε 1) u 2(r 2 ε 2) u n(r n ε n)c = {}^{u_1}(r_1^{\varepsilon_1}){}^{u_2}(r_2^{\varepsilon_2})\ldots {}^{u_n}(r_n^{\varepsilon_n})

where each ε i\varepsilon_i is +1 + 1 or 1 - 1. One also says such elements are consequences of RR. Heuristically an identity among the relations of 𝒫\mathcal{P} is such an element cc which equals 1.

Transformations of a group presentation

Given a group presentation, it is natural to perform transformations using substitutions, say adding in one new symbol for a string of generators, and adjusting the presentation accordingly. The valid transformations that do not change the group being presented are formalised as the Tietze transformations.

Combinatorial group theory

The study of group presentations, their transformations etc. forms part of combinatorial group theory.

Presentations of monoids and other algebraic and categorical structures.

The theory of group presentations generalises to that of presentations of monoids and then to general rewriting systems.


Textbook accounts:

In the context of combinatorial group theory:

Lecture notes:

  • Derek Holt (notes by Florian Bouyer), Presentation of Groups (2013) [pdf]

On (finitely) presented groups as fundamental groups of (finite) simplicial complexes/CW-complexes:

  • Joseph J. Rotman, around Thm. 7.34 in: An Introduction to Algebraic Topology, Graduate Texts in Mathematics 119 (1988) [[doi:10.1007/978-1-4612-4576-6]]

  • Behrooz Mashayekhy, Hanieh Mirebrahimi, Some Properties of Finitely Presented Groups with Topological Viewpoints, International Journal of Mathematics, Game Theory and Algebra 18 6 (2010) 511-515 [arXiv:1012.1744]

On rewriting in group presentations:

category: group theory

Last revised on April 19, 2023 at 17:47:22. See the history of this page for a list of all contributions to it.