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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2407.15785 |
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| _version_ | 1866917736070774784 |
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| author | Costa, Simone Della Fiore, Stefano |
| author_facet | Costa, Simone Della Fiore, Stefano |
| contents | In this paper, we introduce a weakening of the Freiman isomorphisms between subsets of non necessarily abelian groups.
Inspired by the breakthrough result of Kravitz, [14], on cyclic groups, as a first application, we prove that any subset of size $k$ of the dihedral group $D_{2m}$ (and, more in general, of a class of semidirect products) is sequenceable, provided that the prime factors of $m$ are larger than $k!$. Also, a refined bound of $k!/2$ for the size of the prime factors of $m$ can be obtained for cyclic groups $\mathbb{Z}_m$, slightly improving the result of [14]. Then, applying again the concept of weak Freiman isomorphism, we show that any subset of size $k$ of the dicyclic group $\mathrm{Dic}_{m}$ is sequenceable, provided that the prime factors of $m$ are larger than $k^k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_15785 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Weak Freiman isomorphisms and sequencings of small sets Costa, Simone Della Fiore, Stefano Combinatorics 11B75 In this paper, we introduce a weakening of the Freiman isomorphisms between subsets of non necessarily abelian groups. Inspired by the breakthrough result of Kravitz, [14], on cyclic groups, as a first application, we prove that any subset of size $k$ of the dihedral group $D_{2m}$ (and, more in general, of a class of semidirect products) is sequenceable, provided that the prime factors of $m$ are larger than $k!$. Also, a refined bound of $k!/2$ for the size of the prime factors of $m$ can be obtained for cyclic groups $\mathbb{Z}_m$, slightly improving the result of [14]. Then, applying again the concept of weak Freiman isomorphism, we show that any subset of size $k$ of the dicyclic group $\mathrm{Dic}_{m}$ is sequenceable, provided that the prime factors of $m$ are larger than $k^k$. |
| title | Weak Freiman isomorphisms and sequencings of small sets |
| topic | Combinatorics 11B75 |
| url | https://arxiv.org/abs/2407.15785 |