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Quotients of Subdirect Products and Regular p-Groups (Kourovka Notebook Problem 13.19)

Year: 2026
Type: preprint
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. E. I. Khukhro has checked the solution (personal communication; version 1.1). This is an unrefereed note. Version 1.1 (18 October 2026): E. I. Khukhro checked the solution of Problem 13.19 given in the note, including Theorem 4.2 (personal communication), and his two remarks are added as Remark 4.3 (free nilpotent groups of larger class can be used as well; the inclusion used in the proof of Theorem 4.2 is known in folklore as Higman's lemma, with a reference to his book). No theorem and no proof changed. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: KOU-13.19 (Kourovka Notebook, Problem 13.19, Yu. M. Gorchakov, 13th issue, 1995).
DOI: 10.5281/zenodo.23445975
Language: English
Open Access status: green
OpenAlex ID: W7222884842
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