If we then let "N" be the subgroup of "F" generated by all conjugates "x"−1"Rx" of "R", then it follows by definition that every element of "N" is a finite product "x"1−1"r"1"x"1 formula_5 "xm"−1"rm" "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  All three presentations are equivalent. 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:  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 formula_9. To form a group with presentation formula_10, the idea is to take formula_9 quotient by the smallest normal subgroup such that each element of "R" gets identified with the identity. Note that "R" might not be a subgroup, let alone a normal subgroup of formula_9 , so we cannot take a quotient by "R". The solution is to take the normal closure "N" of "R" in formula_9 . The group formula_10 is then defined as the quotient group The elements of "S" are called the generators of formula_10 and the elements of "R" are called the relators. A group "G" is said to have the presentation formula_10 if "G" is isomorphic to formula_10. It is a common practice to write relators in the form formula_19 where "x" and "y" are words on "S". What this means is that formula_20. This has the intuitive meaning that the images of "x" and "y" are supposed to be equal in the quotient group.