The quaternion group as a subgroup of the sphere braid groups

Daciberg Lima Gonçalves (1) , John Guaschi (2)

1 IME - Instituto de Matemática e Estatística
2 LEP - Laboratoire Émile Picard

Let n be greater than or equal to 3. We prove that the quaternion group of order 8 is realised as a subgroup of the sphere braid group B_n(S^2) if and only if n is even. If n is divisible by 4 then the commutator subgroup of B_n(S^2) contains such a subgroup. Further, for all n greater than or equal to 3, B_n(S^2) contains a subgroup isomorphic to the dicyclic group of order 4n.

Mots clés: sphere braid group; quaternion group; dicyclic group; torsion element

Domaines: Topologie géométrique [math.GT]; Théorie des groupes [math.GR]

Daciberg Lima Gonçalves, John Guaschi. The quaternion group as a subgroup of the sphere braid groups. Bulletin of the London Mathematical Society, 2007, 39 (2), pp.232-234. ⟨10.1112/blms/bdl041⟩. ⟨hal-00020804⟩