Quotients of the Artin braid groups and crystallographic groups

Abstract: Let n be greater than or equal to 3. We study the quotient group B_n/[P n,P_n] of the Artin braid group B_n by the commutator subgroup of its pure Artin braid group P_n. We show that B_n/[P n,P_n] is a crystallographic group, and in the case n=3, we analyse explicitly some of its subgroups. We also prove that B_n/[P n,P_n] possesses torsion, and we show that there is a one-to-one correspondence between the conjugacy classes of the finite-order elements of B_n/[P n,P_n] with the conjugacy classes of the elements of odd order of the symmetric group S_n, and that the isomorphism class of any Abelian subgroup of odd order of S_n is realised by a subgroup of B_n/[P n,P_n]. Finally, we discuss the realisation of non-Abelian subgroups of S_n of odd order as subgroups of B_n/[P n,P_n], and we show that the Frobenius group of order 21, which is the smallest non-Abelian group of odd order, embeds in B_n/[P n,P_n] for all n greater than or equal to 7.