In mathematics, an Artin group (or generalized braid group) is a group with a presentation of the form

<math>\left\langle x_1,x_2,\ldots,x_n|\langle x_1, x_2 \rangle^{m_{1,2}}=\langle x_2, x_1 \rangle^{m_{1,2}}, \langle x_1, x_3 \rangle^{m_{1,3}}=\langle x_3, x_1 \rangle^{m_{1,3}}, \ldots \langle x_{n-1}, x_n \rangle^{m_{n-1,n}}=\langle x_{n-1}, x_n \rangle^{m_{n,n-1}}\right\rangle<math>

where

<math>m_{i,j} \in {0,1,\ldots, \infty}<math>.

For <math>m < \infty<math>, <math>\langle x_i, x_j \rangle^m<math> represents an alternating product of <math>x_i<math> and <math>x_j<math> of length <math>m<math>, beginning with <math>x_i<math>. (For example, <math>\langle x_i, x_j \rangle^3 = x_ix_jx_i<math> and <math>\langle x_i, x_j \rangle^4 = x_ix_jx_ix_j<math>.) If <math>m=\infty<math>, then there is no relation for <math>x_i<math> and <math>x_j<math>.

The <math>m_{i,j}<math> can be organized into a symmetric matrix, known as the Coxeter matrix of the group. Each Artin group has as a quotient the Coxeter group with the same set of generators and Coxeter matrix. The kernel of the homomorphism to the associated Coxeter group is generated by relations of the form <math>{x_i}^2=1<math>.

Braid groups are examples of Artin groups, with Coxeter matrix <math>m_{i,i+1} = 3<math> and <math>m_{i,j}=2<math> for <math>|i-j|>1<math>