Complete decomposition of the generalized quaternion groups
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}|...
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| Format: | Article |
| Language: | English |
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De Gruyter
2023-09-01
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| Series: | Open Mathematics |
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| Online Access: | https://doi.org/10.1515/math-2023-0111 |
| _version_ | 1797688799692587008 |
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| author | Chen Huey Voon Sin Chang Seng |
| author_facet | Chen Huey Voon Sin Chang Seng |
| author_sort | Chen Huey Voon |
| collection | DOAJ |
| description | 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}. |
| first_indexed | 2024-03-12T01:35:55Z |
| format | Article |
| id | doaj.art-233e6869770b47d4921d2b8954680b3d |
| institution | Directory Open Access Journal |
| issn | 2391-5455 |
| language | English |
| last_indexed | 2024-03-12T01:35:55Z |
| publishDate | 2023-09-01 |
| publisher | De Gruyter |
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| spelling | doaj.art-233e6869770b47d4921d2b8954680b3d2023-09-11T07:00:02ZengDe GruyterOpen Mathematics2391-54552023-09-01211396310.1515/math-2023-0111Complete decomposition of the generalized quaternion groupsChen Huey Voon0Sin Chang Seng1Mathematical and Actuarial Sciences, Universiti Tunku Abdul Rahman, Kajang, Selangor, MalaysiaSchool of Foundation Studies, Xiamen University, Sepang, MalaysiaLet 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}.https://doi.org/10.1515/math-2023-0111generalized quaternion groupscomplete decompositionsubsets product05a1820d60 |
| spellingShingle | Chen Huey Voon Sin Chang Seng Complete decomposition of the generalized quaternion groups Open Mathematics generalized quaternion groups complete decomposition subsets product 05a18 20d60 |
| title | Complete decomposition of the generalized quaternion groups |
| title_full | Complete decomposition of the generalized quaternion groups |
| title_fullStr | Complete decomposition of the generalized quaternion groups |
| title_full_unstemmed | Complete decomposition of the generalized quaternion groups |
| title_short | Complete decomposition of the generalized quaternion groups |
| title_sort | complete decomposition of the generalized quaternion groups |
| topic | generalized quaternion groups complete decomposition subsets product 05a18 20d60 |
| url | https://doi.org/10.1515/math-2023-0111 |
| work_keys_str_mv | AT chenhueyvoon completedecompositionofthegeneralizedquaterniongroups AT sinchangseng completedecompositionofthegeneralizedquaterniongroups |