Presentation of A Group - Background

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 R x of R, then it is straightforward to show that every element of N is a finite product x₁ −1 r₁ x₁ . . . 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