| Summary: | Let GG be a finite nonabelian group. For any integer m≥2m\ge 2, let A1,…,Am{A}_{1},\ldots ,{A}_{m} be nonempty subsets of GG. If A1,…,Am{A}_{1},\ldots ,{A}_{m} are mutually disjoint and if the subset product A1…Am={α1…αm∣αv∈Av,v=1,2,…,m}{A}_{1}\ldots {A}_{m}=\left\{{\alpha }_{1}\ldots {\alpha }_{m}| {\alpha }_{v}\in {A}_{v},v=1,2,\ldots ,m\right\} coincides with GG, then (A1,…,Am)\left({A}_{1},\ldots ,{A}_{m}) is called a complete decomposition of GG of order mm. In this article, we let GG be the generalized quaternion groups Q2n{Q}_{{2}^{n}}, which is a finite nonabelian group of order 2n{2}^{n} with group presentation given by ⟨x,y∣x2n−1=1,y2=x2n−2,yx=x2n−1−1y⟩\langle x,y| {x}^{{2}^{n-1}}=1,{y}^{2}={x}^{{2}^{n-2}},yx={x}^{{2}^{n-1}-1}y\rangle for positive integer n≥3n\ge 3. We determine the existence of complete decompositions of Q2n{Q}_{{2}^{n}} of order kk, for k∈{2,3,…,2n−1}k\in \left\{2,3,\ldots ,{2}^{n-1}\right\}, and show that Q2n{Q}_{{2}^{n}} can be written in the product of 2n−1{2}^{n-1} subsets, i.e., Q2n=A1A2⋯A2n−1{Q}_{{2}^{n}}={A}_{1}{A}_{2}\cdots {A}_{{2}^{n-1}}, where ∣Aj∣=2| {A}_{j}| =2, for j∈{1,2,…,2n−1}j\in \left\{1,2,\ldots ,{2}^{n-1}\right\}. In addition, we construct the non-complete decomposition of Q2n{Q}_{{2}^{n}} of order kk, for k∈{2,3,…,2n−1}k\in \left\{2,3,\ldots ,{2}^{n-1}\right\}, using the non-exhaustive subsets of Q2n{Q}_{2n}.
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