Alternating groups as products of four conjugacy classes

TL;DR

The paper proves that for large enough alternating groups, the entire group can be expressed as a product of four large normal subsets, each exceeding a specific size threshold, highlighting a product decomposition property.

Contribution

It establishes a new threshold for the size of normal subsets needed to generate the entire alternating group through their product.

Findings

For sufficiently large n, four normal subsets of size at least |G|^{1/2+ε} generate G when multiplied.

The result holds for all ε > 0 with an explicit N(ε).

Demonstrates a product decomposition property in large alternating groups.

Abstract

Let $G$ be the alternating group $\mbox{Alt}(n)$ on $n$ letters. We prove that for any $\varepsilon > 0$ there exists $N = N(\varepsilon) \in \mathbb{N}$ such that whenever $n \geq N$ and $A$, $B$, $C$, $D$ are normal subsets of $G$ each of size at least $|G|^{1/2+\varepsilon}$, then $ABCD = G$.