Alternating groups as products of cycle classes - II

TL;DR

This paper determines the exact maximum size of symmetric groups where every element can be expressed as a product of k l-cycles, revealing the Herzog-Kaplan-Lev conjecture is false for k ≥ 5 and showing the difference grows linearly with k.

Contribution

It provides the exact values of n(k,l) for cases where 3 does not divide l and k ≥ 5, disproving the conjecture and generalizing previous results.

Findings

Exact values of n(k,l) for 3∤l and k≥5 are established.

The Herzog-Kaplan-Lev conjecture is shown to be false for k≥5.

The difference between actual and conjectured n(k,l) grows linearly with k.

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 conjectured that $\lfloor \frac{2kl}{3} \rfloor \leq n(k,l)\leq \lfloor \frac{2kl}{3}\rfloor+1$. It is known that the conjecture holds when $k=2,3,4$. Moreover, it is also true when $3\mid l$. In this article, we determine the exact value of $n(k,l)$ when $3\nmid l$ and $k\geq 5$. As an immediate consequence, we get that $n(k,l)<\lfloor \frac{2kl}{3}\rfloor$ when $k\geq 5$, which shows that the above conjecture is not true in general. In fact, the difference between the exact value of $n(k,l)$ and the conjectured value grows linearly in terms of $k$. Our results also generalize the case of $k=2,3,4$.