text: However, we have, for example, , , etc., so such products are not unique in D8. Each such product equivalence can be expressed as an equality to the identity, such as
  ,
  , or
  .
  Informally, we can consider these products on the left hand side as being elements of the free group , and let .  That is, we let R be the subgroup generated by the strings rfrf, r8, f2, each of which is also equivalent to 1 when considered as products in D8.
  If we then let N be the subgroup of F generated by all conjugates x−1Rx of R, then it follows by definition that every element of N is a finite product x1−1r1x1 ... xm−1rm xm of members of such conjugates. It follows that each element of N, when considered as a product in D8, will also evaluate to 1; and thus that N is a normal subgroup of F. Thus D8 is isomorphic to the quotient group . We then say that D8 has presentation
  Here the set of generators is }, and the set of relations is .  We often see R abbreviated, giving the presentation
  An even shorter form drops the equality and identity signs, to list just the set of relators, which is .  Doing this gives the presentation