Braid groups

In this thesis, we decribe the Braid groups of knitters with their genarators and their relations and prove the Artin theorem. We also studied the complex representations of these groups for small indexed Braid groups. Braid groups, one of the many groups in mathematics, emerging, as related to symmetric groups, are a family of non- abelian groups with infinitely many elements.These groups also found application in physics, especially in quantum theory. Description of these groups through representations has been the subject of many problems. Many of these problems are still open problems in mathematics. In this thesis, we tried to decribe these groups by Artin theorem and also we studied two dimentioal matrix representations of the Braid group B_3. We also touched upon other important issues such as the center of the Artin group, the pure Braid group, and the Burau representation. Keywords: Artin Theorem, Complex representations of B_3, Burau representation