# Malle's Conjecture for $S_n\times A$ for $n = 3,4,5$

## TL;DR

This paper develops a framework to prove Malle's conjecture for certain composite number fields, confirming it for specific cases involving symmetric and abelian groups, and providing new insights into known counterexamples.

## Contribution

It introduces a new method to prove Malle's conjecture for $S_n imes A$ over any number field for specific $n$ and abelian groups, expanding the class of proven cases.

## Key findings

- Proves Malle's conjecture for $S_3 imes A$ with $A$ of order coprime to 2.
- Establishes validity for $S_4 imes A$ with $A$ of order coprime to 6.
- Confirms Malle's conjecture for $S_5 imes A$ with $A$ of order coprime to 30.

## Abstract

We propose a framework to prove Malle's conjecture for the compositum of two number fields based on proven results of Malle's conjecture and good uniformity estimates. Using this method we can prove Malle's conjecture for $S_n\times A$ over any number field $k$ for $n=3$ with $A$ an abelian group of order relatively prime to 2, for $n= 4$ with $A$ an abelian group of order relatively prime to 6 and for $n=5$ with $A$ an abelian group of order relatively prime to 30. As a consequence, we prove that Malle's conjecture is true for $C_3\wr C_2$ in its $S_9$ representation, whereas its $S_6$ representation is the first counter example of Malle's conjecture given by Kl\"uners.