text: The first systematic study was given by Walther von Dyck, student of Felix Klein, in the early 1880s, laying the foundations for combinatorial group theory.
  Examples
  The following table lists some examples of presentations for commonly studied groups. Note that in each case there are many other presentations that are possible. The presentation listed is not necessarily the most efficient one possible.
  An example of a finitely generated group that is not finitely presented is the wreath product  of the group of integers with itself.
  Some theorems
  Theorem. Every group has a presentation.
  To see this, given a group G, consider the free group FG on G. By the universal property of free groups, there exists a unique group homomorphism  whose restriction to G is the identity map. Let K be the kernel of this homomorphism. Then K is normal in FG, therefore is equal to its normal closure, so . Since the identity map is surjective, φ is also surjective, so by the First Isomorphism Theorem, . This presentation may be highly inefficient if both G and K are much larger than necessary.
  Corollary. Every finite group has a finite presentation.
  One may take the elements of the group for generators and the Cayley table for relations.