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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2505.08887 |
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| _version_ | 1866909609576366080 |
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| author | Benavides, Fernando Andres Mutis, Wilson Fernando |
| author_facet | Benavides, Fernando Andres Mutis, Wilson Fernando |
| contents | Given a finite group $G$ and positive integers $r$ and $s$, a problem of interest in algebra is determining the minimum cardinality of the product set $AB$, where $A$ and $B$ are subsets of $G$ such that $|A|=r$ and $|B|=s$. This problem has been solved for the class of abelian groups; however, it remains open for finite non-abelian groups. In this paper, we prove that the result obtained for abelian groups can be extended to the class of metacyclic groups $K_{m,n}=\left\langle a,b \ : \ a^m=1,b^{2n}=a^g,bab^{-1}=a^{-1}\right\rangle$. Consequently, we provide a new proof of the result for the dihedral group $D_n$ and dicylic group $Q_{4n}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_08887 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Minimal Product Set in Non-Abelian Metacyclic Groups of Even Order Benavides, Fernando Andres Mutis, Wilson Fernando Group Theory 11B75, 20D60, 20K01 Given a finite group $G$ and positive integers $r$ and $s$, a problem of interest in algebra is determining the minimum cardinality of the product set $AB$, where $A$ and $B$ are subsets of $G$ such that $|A|=r$ and $|B|=s$. This problem has been solved for the class of abelian groups; however, it remains open for finite non-abelian groups. In this paper, we prove that the result obtained for abelian groups can be extended to the class of metacyclic groups $K_{m,n}=\left\langle a,b \ : \ a^m=1,b^{2n}=a^g,bab^{-1}=a^{-1}\right\rangle$. Consequently, we provide a new proof of the result for the dihedral group $D_n$ and dicylic group $Q_{4n}$. |
| title | Minimal Product Set in Non-Abelian Metacyclic Groups of Even Order |
| topic | Group Theory 11B75, 20D60, 20K01 |
| url | https://arxiv.org/abs/2505.08887 |