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, . . . , σn−1 | σiσj = σjσi, |i − j| > 2; σiσi+1σi = σi+1σiσi+1, 1 ≤ i ≤ n − 2〉. The action of σi on G n is defined by σi(a1, . . . , an) = (a1, . . . , ai−1, ai+1, a ai+1 i , ai+2, . . . , an), where (a1, . . . , an) ∈ G . Note that σ i (a1, . . . , an) = (a1, . . . , ai−1, ai+1, ai, ai+2, . . . , an). An action by σi or σ −1 i on G n is called a Hurwitz move. Two tuples (a1, . . . , an), (b1, . . . , bn) ∈ G n are called (Hurwitz) equivalent, denoted as (a1, . . . , an) ∼ (b1, . . . , bn), if they are in the same Bn-orbit. The (Hurwitz) equivalence class of (a1, . . . , an) ∈ G , i.e., the Bn-orbit of (a1, . . . , an), is denoted by [a1, . . . , an]. If G is a nonabelian group, in general, the Bn-orbits in G n are not known. In [1], Ben-Itzhak and Teicher determined all Bn-orbits in S n m represented by (t1, . . . , tn), where Sm is the symmetric group, each ti is a transposition and t1 · · · tn = 1. It is obvious that if the electronic journal of combinatorics 15 (2008), #R80 1 a1, . . . , an ∈ G generate a finite subgroup, then the Bn-orbit of (a1, . . . , an) in G n is finite. It has been proved that if s1, . . . , sn ∈ GL(R ) are reflections such that the Bn-orbit of (s1, . . . , sn) is finite, then the group generated by s1, . . . , sn is finite; see [2] and [3]. It is natural to ask which types of nonabelian group G allow complete determination of the Bn-orbits in G . In this paper, we show that when G is the generalized quaternion group Q2m or the dihedral group Dpm of order 2p , where p is a prime, the answer to the above question is affirmative. 2 The Generalized Quaternion Group Let m ≥ 2. The generalized quaternion group Q2m of order 2 m is given by the presentation Q2m = 〈α, β | α 2 = 1, α m−2 = β, βαβ = α〉. Each element of Q2m can be uniquely written as α β, where 0 ≤ i < 2 and 0 ≤ j ≤ 1. We have (αβ) β = α l(i−2kj)βj, (2.1) αβ (αβ) = α jk+2ilβl. (2.2) Thus in Qn2m , a Hurwitz move gives one of the following equivalences: (· · · , αβ, αβ, · · · ) ∼ (· · · , αβ, α l(i−2kj)βj, · · · ), (· · · , αβ, αβ, · · · ) ∼ (· · · , α jk+2ilβl, αβ, · · · ). For easier reading, we rewrite the above equivalences, omitting the · · · ’s, with (j, l) = (0, 0), (0, 1), (1, 0) and (1, 1) respectively. (α, α) ∼ (α, α), (2.3) { (α, αβ) ∼ (αβ, α), (α, αβ) ∼ (αβ, α), (2.4) { (αβ, α) ∼ (α, αβ), (αβ, α) ∼ (α, αβ), (2.5) { (αβ, αβ) ∼ (αβ, αβ) = (αβ, αβ), (αβ, αβ) ∼ (αβ, αβ) = (αβ, αβ). (2.6) Lemma 2.1. (i) (α, αβ) ∼ (α, αβ) for all i, j ∈ Z. (ii) (αβ, αβ) ∼ (αβ, αβ) for all i, j, k ∈ Z. (iii) Let τ, ν, e, f ∈ Z such that 0 ≤ ν ≤ m − 2 and e 6≡ f (mod 2). Then for every g ∈ Z, (α νeβ, α νfβ) ∼ (α ν(e+g)β, α ν(f+g)β). the electronic journal of combinatorics 15 (2008), #R80 2