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

Let \(Q\) be a subgroup of a direct product \(G_1\times\dots\times G_n\) 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 \(n\le p\). 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 \(n\ge p+1\), for instance \(C_p\wr C_p\) for \(n=p+1\); so the conjecture fails as well. Our smallest example has \(p=2\), \(n=3\), \(|Q|=512\), \(|G_i|=32\) and \(Q/H\cong D_8\). 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 \(n\le p\) for every prime, in particular for the odd primes, the two realization theorems and the explicit examples. This is an unrefereed note.