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| Main Author: | |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2311.07061 |
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| _version_ | 1866910341225512960 |
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| author | Kabenyuk, Mikhail |
| author_facet | Kabenyuk, Mikhail |
| contents | Let $G$ be a group. The subsets $A_1,\ldots,A_k$ of $G$ form a complete factorization of group $G$ if if they are pairwise disjoint and each element $g\in G$ is uniquely represented as $g=a_1\ldots a_k$, with $a_i\in A_i$. We prove the following theorem: Let $G$ be a finite nilpotent group. If $|G|=m_1\ldots m_k$ where $m_1,\ldots,m_k$ are integers greater $1$ and $k\geq3$, then there exist subsets $A_1,\ldots,A_k$ of $G$ which form a complete factorization of group $G$ and $|A_i|=m_i$ for all $i=1,2,\ldots,k$. In addition, we give several examples of building complete factorization for some groups and formulate one open question. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_07061 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Complete factorizations of finite groups Kabenyuk, Mikhail Group Theory 20D60 Let $G$ be a group. The subsets $A_1,\ldots,A_k$ of $G$ form a complete factorization of group $G$ if if they are pairwise disjoint and each element $g\in G$ is uniquely represented as $g=a_1\ldots a_k$, with $a_i\in A_i$. We prove the following theorem: Let $G$ be a finite nilpotent group. If $|G|=m_1\ldots m_k$ where $m_1,\ldots,m_k$ are integers greater $1$ and $k\geq3$, then there exist subsets $A_1,\ldots,A_k$ of $G$ which form a complete factorization of group $G$ and $|A_i|=m_i$ for all $i=1,2,\ldots,k$. In addition, we give several examples of building complete factorization for some groups and formulate one open question. |
| title | Complete factorizations of finite groups |
| topic | Group Theory 20D60 |
| url | https://arxiv.org/abs/2311.07061 |