A braid determines a permutation by the rule the i th hole on the first plane z label of the corresponding i th hole on the second plane. This is a surjective homomorphism B_{n} z S_{n} that maps s_{i} to the transposition (i, i+1). This is not an isomorphism because B_{n} is infinite. s_{i} has infinite order and the transposition (i, i+1) squares to the identity (and is finite).

In his 1925 paper “Theorie der Zöpfe ” Artin showed that multiplication in B_{n} is completely described by the relations:

s_{i}s_{j} = s_{j}s_{i} (|i-j| ≥ 2),

s_{i}s_{i+1}s_{i} = s_{i+1}s_{i}s_{i+1}.

In physics the Artin relations are called the Yang-Baxter equations. They are important in statistical physics.

Artin also devised the method of “Combing the Braid” which allowed for the identification of the identity element from an arbitrary word in the generators. An alternative algebraic method by Garside (1967) also decides when two elements are conjugate.

In 2001 a proof by Bigelow and independently by Kramer proved that braid groups are linear.

The braid groups described here are braid groups of the plane, because we started with the two parallel planes being punctured. This method may be generalized to more general curves than the plane.