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 si are elements of S, adjacent si are distinct, and ai 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 . . . xm −1 rm xm 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
Read more about this topic: Presentation Of A Group
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