The Exhaustion Numbers of the Generalized Quaternion Groups

Let G be a finite group and let T be a non-empty subset of G. For any positive integer k, let Tk={t1…tk∣t1,…,tk∈T}. The set T is called exhaustive if Tn=G for some positive integer n where the smallest positive integer n, if it exists, such that Tn=G is called the exhaustion number of T and is denoted by e(T). If Tk≠G for any positive integer k, then T is a non-exhaustive subset and we write e(T)=∞. In this paper, we investigate the exhaustion numbers of subsets of the generalized quaternion group Q2n=⟨x, y∣x2n−1=1, x2n−2=y2, yx=x2n−1−1y⟩ where n≥3. We show that Q2n has no exhaustive subsets of size 2 and that the smallest positive integer k such that any subset T⊆Q2n of size greater than or equal to k is exhaustive is 2n−1+1. We also show that for any integer k∈{3,…,2n}, there exists an exhaustive subset T of Q2n such that |T|=k.