{
  "abstract": "We study isomorphism decision and basis construction for finite Abelian groups of known order $n$. We measure complexity by the number of additions performed in the groups and by the total running time. We give a randomized isomorphism decision algorithm that performs $\\tilde O(n^{1/4})$ group additions and runs in $\\tilde O(n^{1/4})$ time. This improves the $\\tilde O(\\sqrt n)$ upper bound of Chen and Fu and matches Bshouty $Ω(n^{1/4})$ lower bound up to polylogarithmic factors. We also prove that finding a basis with success probability at least $2/3$ requires $Ω(\\sqrt n)$ group additions in the worst case. This matches the $\\tilde O(\\sqrt n)$ upper bound of Chen and Fu up to polylogarithmic factors. These results separate isomorphism decision from basis construction in finite Abelian groups: their optimal worst-case complexities are $\\tildeΘ(n^{1/4})$ and $\\tildeΘ(\\sqrt n)$, respectivel",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.08280",
  "author_total": 1,
  "authors": [
    {
      "name": "Nader H. Bshouty"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e2xeqbj628e78.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.08280",
  "fulltext_url": null,
  "id": "e2xeqbj628e78",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.08280",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.GR"
    ],
    [
      "openalex",
      "1703|Complexity and Algorithms in Graphs"
    ]
  ],
  "openalex": "W7221029381",
  "pdf_url": "https://arxiv.org/pdf/2610.08280",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e2xeqbj628e78.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.08280",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.08280"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Almost Tight Bounds for Isomorphism Testing and Basis Construction in Finite Abelian Groups",
  "title_text": "Almost Tight Bounds for Isomorphism Testing and Basis Construction in Finite Abelian Groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/e2xeqbj628e78",
  "venue": "arXiv",
  "versions": []
}