Let n ě 3. In this paper, we study the quotient group B n {rP n , P n s of the Artin braid group B n by the commutator subgroup of its pure Artin braid group P n. We show that B n {rP n , P n s is a crystallographic group, and in the case n " 3, we analyse explicitly some of its subgroups. We also prove that B n {rP n , P n s 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 {rP n , P n s 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 {rP n , P n s. Finally, we discuss the realisation of non-Abelian subgroups of S n of odd order as subgroups of B n {rP n , P n s, and we show that the Frobenius group of order 21, which is the smallest non-Abelian group of odd order, embeds in B n {rP n , P n s for all n ě 7.

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In this paper we prove that all pure Artin braid groups P n (n ≥ 3) have the R ∞ property. In order to obtain this result, we analyse the naturally induced morphism Aut (P n ) −→ Aut (Γ 2 (P n )/Γ 3 (P n )) which turns out to factor through a representation ρ : S n+1 −→ Aut (Γ 2 (P n )/Γ 3 (P n )). We can then use representation theory of the symmetric groups to show that any automorphism α of P n acts on the free abelian group Γ 2 (P n )/Γ 3 (P n ) via a matrix with an eigenvalue equal to 1. This allows us to conclude that the Reidemeister number R(α) of α is ∞.