Complete decomposition of the generalized quaternion groups - Chen Huey Voon; Sin Chang Seng (Open Mathematics, 2023)
Let GG be a finite nonabelian group. For any integer m≥2m\ge 2, let A1,…,AmA_{1},… ,A_{m} be nonempty subsets of GG. If A1,…,AmA_{1},… ,A_{m} are mutually disjoint and if the subset product A1…Am={α1…αm∣αv∈Av,v=1,2,…,m}{A}_{1}… A_{m}=\left\{{α }_{1}… {α }_{m}| {α }_{v}\in {A}_{v},v=1,2,… ,m\right\} coincides with GG, then (A1,…,Am)\left(A_{1},… ,A_{m}) is called a complete decomposition of GG of order mm. In this article, we let GG be the generalized quaternion groups Q2nQ_{{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 Q2nQ_{{2}^{n}} of order kk, for k∈{2,3,…,2n−1}k\in \left\{2,3,… ,{2}^{n-1}\right\}, and show that Q2nQ_{{2}^{n}} can be written in the product of 2n−1{2}^{n-1} subsets, i.e., Q2n=A1A2⋯A2n−1Q_{{2}^{n}}=A_{1}{A}_{2}… A_{{2}^{n-1}}, where ∣Aj∣=2| A_{j}| =2, for j∈{1,2,…,2n−1}j\in \left\{1,2,… ,{2}^{n-1}\right\}. In addition, we construct the non-complete decomposition of Q2nQ_{{2}^{n}} of order kk, for k∈{2,3,…,2n−1}k\in \left\{2,3,… ,{2}^{n-1}\right\}, using the non-exhaustive subsets of Q2nQ_{2n}.