On Malle's conjecture for the product of symmetric and nilpotent groups

TL;DR

This work proves a strong form of Malle’s conjecture for the product $S_n\times G$ with $n\in\{3,4,5\}$ and $G$ a finite nilpotent group, by embedding $G$ via the regular representation into $S_{|G|}$ and $S_n$ in its natural degree $n$ representation, so that $S_n\times G\hookrightarrow S_{n|G|}$. The authors combine the Klüners–Malle approach with Wang’s results for $S_n$ and a recent nilpotent-extensions parameterization by Koymans–Pagano to establish sharp local uniformity bounds, and they introduce a discriminant-truncation framework $\mathrm{Disc}_Y$ to compare $N_k(S_n\times G; X)$ with the $Y$-localized count $N_{k,Y}(S_n\times G; X)$. A key technical innovation is a tight product-bound principle (extending a lemma for products of two groups) together with a uniformity estimate for nilpotent groups, which together yield the main asymptotic $N_k(S_n\times G; X)\sim c(k,S_n\times G) X^{1/|G|}$ under mild conditions on $G$ (e.g., obstructions only for primes dividing $|G|$ and orders of small primes). The result extends Wang’s earlier work on $S_n\times A$ (with $A$ abelian) and builds a framework likely to apply to broader group families, highlighting the utility of combining local parameterizations with discriminant-control techniques in establishing precise asymptotics for composite Galois groups.

Abstract

Let $G$ be a finite nilpotent group and $n\in \{3,4, 5\}$. Consider $S_n\times G$ as a subgroup of $S_n\times S_{|G|}\subset S_{n|G|}$, where $G$ embeds into the second factor of $S_n\times S_{|G|}$ via the regular representation. Over any number field $k$, we prove the strong form of Malle's conjecture for $S_n\times G$ viewed as a subgroup of $S_{n|G|}$. Our result requires that $G$ satisfies some mild conditions.

Theorems & Definitions (58)

• Conjecture 1.1
• Conjecture 1.2
• Theorem 1.3
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4