On minimal product-one sequences of maximal length over the non-abelian group of order $pq$
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| Format: | Preprint |
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2025
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| _version_ | 1866916984666456064 |
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| author | Avelar, Danilo Vilela Martínez, Fabio Enrique Brochero Ribas, Sávio |
| author_facet | Avelar, Danilo Vilela Martínez, Fabio Enrique Brochero Ribas, Sávio |
| contents | Let $G$ be a finite group. A sequence over $G$ is a finite multiset of elements of $G$, and it is called product-one if its terms can be ordered so that their product is the identity of $G$. The large Davenport constant $\D(G)$ is the maximal length of a minimal product-one sequence, that is, a product-one sequence that cannot be partitioned into two nontrivial product-one subsequences. Let $p,q$ be odd prime numbers with $p \mid q-1$ and let $C_q \rtimes C_p$ denote the non-abelian group of order $pq$. It is known that $\D(C_q \rtimes C_p) = 2q$. In this paper, we describe all minimal product-one sequences of length $2q$ over $C_q \rtimes C_p$. As an application, we further investigate the $k$-th elasticity (and, consequently, the union of sets containing $k$) of the monoid of product-one sequences over these groups. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_00070 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On minimal product-one sequences of maximal length over the non-abelian group of order $pq$ Avelar, Danilo Vilela Martínez, Fabio Enrique Brochero Ribas, Sávio Group Theory Combinatorics Number Theory Let $G$ be a finite group. A sequence over $G$ is a finite multiset of elements of $G$, and it is called product-one if its terms can be ordered so that their product is the identity of $G$. The large Davenport constant $\D(G)$ is the maximal length of a minimal product-one sequence, that is, a product-one sequence that cannot be partitioned into two nontrivial product-one subsequences. Let $p,q$ be odd prime numbers with $p \mid q-1$ and let $C_q \rtimes C_p$ denote the non-abelian group of order $pq$. It is known that $\D(C_q \rtimes C_p) = 2q$. In this paper, we describe all minimal product-one sequences of length $2q$ over $C_q \rtimes C_p$. As an application, we further investigate the $k$-th elasticity (and, consequently, the union of sets containing $k$) of the monoid of product-one sequences over these groups. |
| title | On minimal product-one sequences of maximal length over the non-abelian group of order $pq$ |
| topic | Group Theory Combinatorics Number Theory |
| url | https://arxiv.org/abs/2510.00070 |