A quotient of the Artin braid groups related to crystallographic groups

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• 作者：Daciberg Lima Gonç ; alves^a ; ^{dlgoncal@ime.usp.br} ; John Guaschi^b ; ^{john.guaschi@unicaen.fr} ; Oscar Ocampo^c ; ^{oscaro@ufba.br}
• 关键词：Braid groups ; Crystallographic groups
• 刊名：Journal of Algebra
• 出版年：2017
• 出版时间：15 March 2017
• 年：2017
• 卷：474
• 期：Complete
• 页码：393-423
• 全文大小：602 K
• 卷排序：474

文摘

Let n≥3n≥3. In this paper, we study the quotient group Bn/[Pn,Pn]Bn/[Pn,Pn] of the Artin braid group BnBn by the commutator subgroup of its pure Artin braid group PnPn. We show that Bn/[Pn,Pn]Bn/[Pn,Pn] is a crystallographic group, and in the case n=3n=3, we analyse explicitly some of its subgroups. We also prove that Bn/[Pn,Pn]Bn/[Pn,Pn] possesses torsion, and we show that there is a one-to-one correspondence between the conjugacy classes of the finite-order elements of Bn/[Pn,Pn]Bn/[Pn,Pn] with the conjugacy classes of the elements of odd order of the symmetric group SnSn, and that the isomorphism class of any Abelian subgroup of odd order of SnSn is realised by a subgroup of Bn/[Pn,Pn]Bn/[Pn,Pn]. Finally, we discuss the realisation of non-Abelian subgroups of SnSn of odd order as subgroups of Bn/[Pn,Pn]Bn/[Pn,Pn], and we show that the Frobenius group of order 21, which is the smallest non-Abelian group of odd order, embeds in Bn/[Pn,Pn]Bn/[Pn,Pn] for all n≥7n≥7.