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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 $$n\geq7$$.

##### MSC:
 20F36 Braid groups; Artin groups 20H15 Other geometric groups, including crystallographic groups 20F45 Engel conditions 20E07 Subgroup theorems; subgroup growth
##### Keywords:
braid groups; crystallographic groups
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##### References:
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