Presentations of groups

Idea

In trying to study a group $G$, 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 $S_3$, or, isomorphically, the dihedral group (of symmetries of a triangle) which we will call $D_3$ (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 $a$ and the transposition $b$, we have

What about the other words: $\mathrm{ba}$ for instance. If one calculates $ba$ in $S_3$ and then looks up what you get you have $\mathrm{ba}=a^{2}b$, so that there is a relation between these two words. This however is not all. What about $babaabaaabbaa$ that is a word in the $a$s and $b$s so should represent something in $S_3$, in fact if you thing in the geometric version of $D_3$ 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 $X$, so there will be an epimorphism from $F(X)$ to $G$, and then you try to find a description for the kernel of that epimorphism, which we will denote by $N$. The description of this normal subgroup $N$ is as the normal closure of a set $R$ of relations, i.e. words in the generators or, equally validly, elements in the free group on $X$.

Definition

A presentation of a group $G$ is a pair of sets, written $⟨X:R⟩$ such that for $F=F(X)=⟨X⟩=⟨X:\varnothing ⟩$ is the free group on the set of letters $X$ and $N$ the normal closure of the set of relators $R$, there is a specified isomorphism from $F/N$ to $G$.

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

Sometimes it is convenient to proceed otherwise, however, and to give a specific function from a set $X$ to the set of elements of $G$. This function then induces a group homomorphism from $F=⟨X⟩$ to $G$, and if this is a surjection, then we can find some $N$ (generators for its kernel) to produce a presentation of $G$.

Examples

• $G$ a cyclic group of order $n$ has presentation $⟨a:a^n⟩$.

• $S_3$ has a presentation $⟨a,b:a^3,b^2,(\mathrm{ab}{)}^2⟩$.

• The trefoil knot group has two useful presentations:

  • $⟨a,b:a^3=b^2⟩$, which displays the fact hat the trefoil is a (2,3)-torus knot; and

  • $⟨x,y:\mathrm{xyx}=\mathrm{yxy}⟩$, which shows the link between this group and the Artin braid group, $\mathrm{Br}3$.

Discussion

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

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

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

where each ${\epsilon }_i$ is $+1$ or $-1$. One also says such elements are consequences of $R$. Heuristically an identity among the relations? of $𝒫$ is such an element $c$ 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,

• N. D. Gilbert and T. Porter, Knots and Surfaces, Oxford U.P., 1994.

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:

• Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Springer, 1977; toc.