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(Reference retrieved automatically from Web of Science through information on FAPESP grant and its corresponding number as mentioned in the publication by the authors.)

A quotient of the Artin braid groups related to crystallographic groups

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Author(s):
Goncalves, Daciberg Lima ; Guaschi, John ; Ocampo, Oscar
Total Authors: 3
Document type: Journal article
Source: Journal of Algebra; v. 474, p. 393-423, MAR 15 2017.
Web of Science Citations: 2
Abstract

Let n >= 3. In this paper, we study the quotient group B-n/{[}P-n,P-n] of the Artin braid group Bn 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 Bn/{[}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 Sn is realised by a subgroup of B-n/{[}p(n),p(n)] Finally, we discuss the realisation of non-Abelian subgroups of Sn 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 >= 7. (C) 2016 Elsevier Inc. All rights reserved. (AU)