The classification and the conjugacy classes of the finite subgroups of the sphere braid groups
Daciberg Lima Gon\c{c}alves (IME), John Guaschi (LMNO)

TL;DR
This paper classifies the finite subgroups of sphere braid groups B_n(S^2), identifying maximal groups, their conjugacy classes, and their geometric and algebraic constructions, extending understanding of torsion elements and subgroup structure.
Contribution
It provides a complete classification of finite subgroups of B_n(S^2), including explicit constructions and conjugacy class counts, linking to torsion elements and series of the group.
Findings
Maximal finite subgroups identified for various n
Explicit algebraic and geometric constructions provided
Number of conjugacy classes of finite subgroups determined
Abstract
Let n\geq 3. We classify the finite groups which are realised as subgroups of the sphere braid group B_n(S^2). Such groups must be of cohomological period 2 or 4. Depending on the value of n, we show that the following are the maximal finite subgroups of B_n(S^2): Z_{2(n-1)}; the dicyclic groups of order 4n and 4(n-2); the binary tetrahedral group T_1; the binary octahedral group O_1; and the binary icosahedral group I. We give geometric as well as some explicit algebraic constructions of these groups in B_n(S^2), and determine the number of conjugacy classes of such finite subgroups. We also reprove Murasugi's classification of the torsion elements of B_n(S^2), and explain how the finite subgroups of B_n(S^2) are related to this classification, as well as to the lower central and derived series of B_n(S^2).
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