Alternating groups as products of cycle classes

TL;DR

This paper determines the maximum size of symmetric groups where every element can be expressed as a product of fixed-length cycles, confirming a conjecture and providing new exact values for specific cases.

Contribution

The paper proves a conjecture about the maximum size of alternating groups expressible as products of cycles and computes new exact values for particular parameters.

Findings

Confirmed Herzog-Kaplan-Lev conjecture for even k and divisible by 3 l

Established that n(k,3)=2k+1 for odd k

Extended understanding of cycle product decompositions in alternating groups

Abstract

Given integers $k,l\geq 2$, where either $l$ is odd or $k$ is even, let $n(k,l)$ denote the largest integer $n$ such that each element of $A_n$ is a product of $k$ many $l$-cycles. In 2008, M. Herzog, G. Kaplan and A. Lev proved that if $k,l$ both are odd, $3\mid l$ and $l>3$, then $n(k,l)=\frac{2}{3}kl$. They further conjectured that if $k$ is even and $3\mid l$, then $n(k,l)=\frac{2}{3}kl+1$. In this article, we prove this conjecture. We also prove that $n(k,3)=2k+1$ if $k$ is odd.