text: All three presentations are equivalent.
  Notation
  Although the notation   used in this article for a presentation is now the most common, earlier writers used different variations on the same format.  Such notations include the following:
  }
  Definition
  Let S be a set and let FS be the free group on S.  Let R be a set of words on S, so R naturally gives a subset of . To form a group with presentation , take the quotient of  by the smallest normal subgroup that contains each element of R.  (This subgroup is called the normal closure N of R in .)  The group  is then defined as the quotient group
  The elements of S are called the generators of  and the elements of R are called the relators. A group G is said to have the presentation  if G is isomorphic to .
  It is a common practice to write relators in the form  where x and y are words on S. What this means is that . This has the intuitive meaning that the images of x and y are supposed to be equal in the quotient group. Thus, for example, rn in the list of relators is equivalent with .