Multiplicative Subgroups of $\mathbb{Z}_p^*$ that are Generalized Arithmetic Progressions
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866908811557601280 |
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| author | Cochrane, Albert |
| author_facet | Cochrane, Albert |
| contents | We prove that a multiplicative subgroup $A_k$ of $\mathbb{Z}_p^*$ is a generalized arithmetic progression if and only if $|A_k| = 2,\ 4,$ or $p-1$. Much of the argument is built upon recent work studying additive decompositions of subgroups of $\mathbb{Z}_p^*$, and we generalize a result of Hanson and Petridis to show that any additive $n$-decomposition of a subgroup must be a direct sum. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_04111 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Multiplicative Subgroups of $\mathbb{Z}_p^*$ that are Generalized Arithmetic Progressions Cochrane, Albert Number Theory 11B30 (Primary) 11B13, 11B25 (Secondary) We prove that a multiplicative subgroup $A_k$ of $\mathbb{Z}_p^*$ is a generalized arithmetic progression if and only if $|A_k| = 2,\ 4,$ or $p-1$. Much of the argument is built upon recent work studying additive decompositions of subgroups of $\mathbb{Z}_p^*$, and we generalize a result of Hanson and Petridis to show that any additive $n$-decomposition of a subgroup must be a direct sum. |
| title | Multiplicative Subgroups of $\mathbb{Z}_p^*$ that are Generalized Arithmetic Progressions |
| topic | Number Theory 11B30 (Primary) 11B13, 11B25 (Secondary) |
| url | https://arxiv.org/abs/2602.04111 |