Presentation of a group - Definition and Overview

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 some of these generators, and a set T of relations among those generators (I.E. the generators of identity words). We then say G has presentation (S;T).

Every group has a presentation, and in fact many; a presentation is often the most compact way of describing the structure of the group. Do not confuse a presentation with a representation, which studies the properties of abstract groups via their representations as linear transformations of vector spaces.

Formally, the group is then specified as being isomorphic to a quotient group of a free group, where the free group is given by S and the quotient group is specified by the relations T.

Background

A free group on a set S = {s_i} is a group where each element can be uniquely described as a finite length product of the form:

x₁^a₁ x₂^a₂ ... x_n^a_n

where each x_i is a member of S, and x_i ≠ x_i+1 for any i; and each a_i is any non-zero integer.

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 D₈ can be generated by a rotation, r, of order 4; 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, r f r = f, r³ = r ^-1, etc.; so such products are not unique in D₈. Each such product equivalence can be expressed as an equality to the identity; such as

r f r f = 1

r ⁴ = 1

f ² = 1

Informally, we can consider these products on the left hand side as being elements of the free group F = F({r,f}), and can consider the subgroup H 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 K be the subgroup of F generated by all conjugates x ^-1 H x of H, then K is a normal subgroup of F; and each element of K, when considered as a product in D₈, will also evaluate to 1.

This leads us to consider the quotient group F/K, and the associated homomorphism U | F → F/K, with K = Ker(U). Then D₈ is isomorphic to F/K; and we write D₈ has presentation ({r,f}; {r⁴, f², (rf)²}).

Formal definition

More formally, let S be a set, and T be a set of words in letters of S such that each t in T is of the form:

x₁^a₁ x₂^a₂ ... x_n^a_n

where each x_i is a member of S, and x_i ≠ x_i+1 for any i; and each a_i is any non-zero integer.

Then if F is freely generated by S, there is an obvious function f | T → F. Let T^F be the (normal) subgroup of F generated by all conjugates of f(T) in F; then if G = F/T^F, we say that G has representation (S;T), or equivalently, that G is generated by S with relations T.

Examples

If G is freely generated by S, then G has presentation (S; {}).

For any natural number n, the cyclic group C_n has presentation ({x}; {xⁿ}).

If D_2n is the dihedral group with 2n elements, then it has presentation ({x,y}; {xⁿ, y², (xy)²}).

The symmetric group S₄ has presentation ({x,y}; {x³, y⁴, (xy)²}).

Some theorems

Every group G has a presentation, in fact infinitely many.

Every finite group has a finite presentation, that is, a presentation <math>(S;T)<math> such that <math>S<math> and <math>T<math> are finite.

The negative solution to the word problem for groups states that there is no general algorithm which, for a given presentation (S;T) and two words u, v, decides whether u and v describe the same element in the group.

Further reading

Group Theory, W. R. Scott, Dover Publications