{
  "schema_version": 1,
  "problem_number": "KOU-13.19",
  "title": "Quotients of Subdirect Products and Regular p-Groups (Kourovka Notebook Problem 13.19)",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let Q be a subgroup of a direct product G_1 × ··· × G_n of groups and H a normal subgroup of Q such that Q and H both project onto every factor. Yu. M. Gorchakov asked in the Kourovka Notebook (Problem 13.19) whether Q/H is a regular p-group whenever it is a p-group; in 2009 he stated this as a conjecture for quotients of class exactly n − 1. We show that Q/H is necessarily regular if and only if n ≤ p. The positive half is classical: Q/H has class at most n − 1 by Gorchakov's lemma of 1965, and p-groups of class less than p are regular by a theorem of P. Hall. For the negative half we show, with Magnus's theorem on free groups, that every nilpotent group of class at most n − 1 is such a quotient with n free factors, and that a finite p-group whose n-th modular dimension subgroup is trivial is such a quotient with n isomorphic finite p-group factors. With Jennings's theorem this gives irregular quotients of class exactly n − 1 for every n ≥ p + 1, for instance C_p ≀ C_p for n = p + 1; so the conjecture fails as well. Our smallest example has p = 2, n = 3, |Q| = 512, |G_i| = 32 and Q/H ≅ D_8. For p = 2 an irregular quotient with four perfect factors can be extracted from a 2018 theorem of Kearnes, Mayr and Ruškuc, who do not discuss regularity; whether examples of D. M. Evans (1993) give one was not determined. What this note adds is the threshold n ≤ p for every prime, in particular for the odd primes, the two realization theorems and the explicit examples. This is an unrefereed note.",
  "result_type": "COMPLETE_NEGATIVE_ANSWER",
  "categories": [
    "math.GR"
  ],
  "keywords": [
    "subdirect products",
    "regular p-groups",
    "nilpotency class",
    "dimension subgroups",
    "Magnus embedding",
    "Jennings's theorem",
    "wreath products",
    "Kourovka Notebook",
    "Problem 13.19",
    "Gorchakov",
    "complete answer",
    "UnsolvedMath",
    "KOU-13.19",
    "math.GR",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-18",
  "publication_date": "2026-10-18",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-18",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/kou-13-19/",
  "pdf_url": "https://eulersolve.org/papers/kou-13-19/paper.pdf?v=d64b3d0666e2",
  "doi": "10.5281/zenodo.23445976",
  "zenodo_record_url": "https://zenodo.org/records/23445976",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The class bound is Gorchakov's lemma of 1965; an irregular quotient for p = 2 is implicit in Kearnes, Mayr and Ruškuc (2018). The note adds the threshold for every prime, two realization theorems and explicit examples. Unrefereed note.",
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    "source.zip": {
      "sha256": "259a79f18142c07fd81ea7303e50978734244cc75301f5ce817b1a0241cd1cb6"
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  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}