Hurwitz Equivalence in Dihedral Groups

Hurwitz Equivalence in Dihedral Groups

In this paper we determine the orbits of the braid group Bn action on Gn when G is a dihedral group and for any T ∈ Gn. We prove that the following invariants serve as necessary and sufficient conditions for Hurwitz equivalence. They are: the product of its entries, the subgroup generated by its entries, and the number of times each conjugacy class (in the subgroup generated by its entries) is ...

Hurwitz Equivalence in Tuples of Dihedral Groups, Dicyclic Groups, and Semidihedral Groups

Let D2N be the dihedral group of order 2N , Dic4M the dicyclic group of order 4M , SD2m the semidihedral group of order 2 m, and M2m the group of order 2 m with presentation M2m = 〈α, β | α 2m−1 = β2 = 1, βαβ−1 = α2 m−2+1〉. We classify the orbits in Dn 2N , Dic n 4M , SD n 2m , and M n 2m under the Hurwitz action.

Hurwitz Equivalence in Tuples of Generalized Quaternion Groups and Dihedral Groups

Let Q2m be the generalized quaternion group of order 2 m and DN the dihedral group of order 2N . We classify the orbits in Q2m and D n pm (p prime) under the Hurwitz action. 1 The Hurwitz Action Let G be a group. For a, b ∈ G, let a = bab and a = bab. The Hurwitz action on G (n ≥ 2) is an action of the n-string braid group Bn on G . Recall that Bn is given by the presentation Bn = 〈σ1, . . . , ...

Sumsets in dihedral groups

Let Dn be the dihedral group of order 2n. For all integers r, s such that 1 ≤ r, s ≤ 2n, we give an explicit upper bound for the minimal size μDn (r, s) = min |A · B| of sumsets (product sets) A · B, where A and B range over all subsets of Dn of cardinality r and s respectively. It is shown by construction that μDn (r, s) is bounded above by the known value of μG (r, s), where G is any abelian ...

Dihedral Groups