Group Presentation

• Let be a set and . Let be the least normal subgroup of containing each . An isomorphism of onto a group is a presentation of .

  and give a group presentation denoted as . The set is the set of generators for the presentation and each is called a relator. Each is a consequence of .

  An equation is a relation.

  A presentation is finite if both and are finite sets.

• If , then the group with the presentation can be denoted as .

• Two presentations are isomorphic if they give isomorphic groups.

• Group presentations present three undecidable problems:

  • Determining if two presentations are isomorphic
  • Determining if a group given by a presentation is finite, free, abelian, or trivial.
  • The word problem — determine if word is a consequence of a given set of relations .

• Every group has a presentation.

  • Follows because every group has a homomorphic image to a free group. For any such mapping we can simply take as .

• The general idea behind Group Presentations is that they allow us to make groups that are as similar to a specified free group as possible subject to certain equations that act as constraints.

Links §

• A First Course in Abstract Algebra 7th Edition by Fraleigh