{
  "annotation_id": "f4018e5f-5f0e-4fa3-b607-a9505981f48d",
  "date_created": "2026-02-17T05:53:11.278000Z",
  "date_modified": "2026-02-17T05:53:11.278000Z",
  "file_hash": "5cb7e7f017c45c23b8400f45a04f7d2db6a6967b9f4795992dfc23a9d3ec286b",
  "private": false,
  "record": {
    "abstract": "In this paper, we characterize the direct sum decompositions of the cyclic group $\\mathbb{Z}_{(pqr)^2}$, where $p$, $q$, and $r$ are distinct primes. We show that if $A \\oplus B = \\mathbb{Z}_{(pqr)^2}$ with $|A| = |B| = pqr$, then Sands\u0027 conjecture fails to hold, in other words, neither $A$ nor $B$ is contained in a proper subgroup of $\\mathbb{Z}_{(pqr)^2}$, if and only if the sets $A, B$ form a Szab\\\u0027{o} pair.",
    "arxiv_id": "2601.07135",
    "authors": [
      "Xin-Rong Dai"
    ],
    "categories": [
      "math.CO"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "title": "Decompositions for Cyclic Groups with 3 Prime Factors",
    "url": "https://arxiv.org/abs/2601.07135",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "f5ab433c-568e-4666-8307-02114dd2590c",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}