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A quotient of the Artin braid groups related to crystallographic groups. (English) Zbl 1367.20034
Authors’ abstract: Let \(n\geq3\). In this paper, 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 \(n\geq7\).

20F36 Braid groups; Artin groups
20H15 Other geometric groups, including crystallographic groups
20F45 Engel conditions
20E07 Subgroup theorems; subgroup growth
Full Text: DOI
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