nLab
group presentation

Presentations of groups

Idea

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,ab,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 thing in the geometric version of 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 possibles.

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.

Definition

A presentation of a group, GG, is a coequalizer diagram FRFXGF R \rightrightarrows F X \twoheadrightarrow G, where FXFX is the free group on a set of generators, and FRFR 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

  • GG a 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.

  • S 3S_3 has a presentation a,b:a 3,b 2,(ab) 2\langle a,b : a^3, b^2,(ab)^2 \rangle.

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

Discussion

  • ‘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.

References

A basic introduction to the theory of group presentations can be found in some standard texts, both on group theory itself and on related areas of low dimensional topology, for instance,

For more elementary or introductory texts, see, for instance,

  • D. L. Johnson, Presentations of Groups (London Mathematical Society Student Texts 15) 1990, Cambridge Univ. Press,

or his earlier:

  • D. L. Johnson, Topics in Theory of Group Presentations, London Mathematical Society Lecture notes series 42) 1980, Cambridge Univ. Press,

which have some very useful material in them.

For a deeper treatment of the area, more specialised sources are needed:

category: group theory

Revised on January 26, 2014 21:09:31 by Tim Porter (2.26.28.54)