The Automorphism Groups of the Braid Groups

In the first of two papers published in the Annals in 1947 [3] Emil Artin mentioned the problem of determining all automorphisms of the braid groups (of the Euclidean plane), and in the second [4] took a first step towards a solution. The main result of this paper is a complete determination of these automorphism groups: the outer automorphism group is of order two, generated by the automorphism class containing the element which inverts each of the standard braid generators. Visually, a plane projection of the automorphic image of a braid m is obtained from one for f by replacing each over-crossing by an under-crossing. This result establishes a recent conjecture of Pietrowski and Solitar, to whom we are indebted for bringing the problem to our attention. They obtained the first determination of the automorphism group of the fourstring braid group, using techniques for computing automorphism groups of amalgamated free products [13]. (The result for braids on at most three strings is easy and seems part of the folklore.) The argument we employ is algebraic, and is patterned after that