The classification and the conjugacy classes of the finite subgroups of the sphere braid groups

Let n 3. We classify the finite groups which are realised as subgroups of the sphere braid group B n .S 2 /.Such groups must be of cohomological period 2 or 4. Depending on the value of n, we show that the following are the maximal finite subgroups of B n .S 2 /: Z 2.n 1/ ; the dicyclic groups of order 4n and 4.n 2/; the binary tetrahedral group T ; the binary octahedral group O ; and the binary icosahedral group I .We give geometric as well as some explicit algebraic constructions of these groups in B n .S 2 / and determine the number of conjugacy classes of such finite subgroups.We also reprove Murasugi's classification of the torsion elements of B n .S 2 / and explain how the finite subgroups of B n .S 2 / are related to this classification, as well as to the lower central and derived series of B n .S 2 /.