The p-width of the alternating groups

TL;DR

This paper investigates the minimal number of elements of prime order needed to express any element in alternating groups, establishing an upper bound of three and demonstrating its sharpness for various primes.

Contribution

It proves that the p-width of alternating groups is at most 3 for all sufficiently large n, and shows this bound is optimal for each prime p.

Findings

The p-width of A_n is at most 3 for all n ≥ p.

The bound of 3 is sharp, with examples achieving this maximum.

The result applies to all primes p and large enough n.

Abstract

Let $p$ be a fixed prime. For a finite group generated by elements of order $p$, the $p$-width is defined to be the minimal $k\in\mathbb{N}$ such that any group element can be written as a product of at most $k$ elements of order $p$. Let $A_{n}$ denote the alternating group of even permutations on $n$ letters. We show that the $p$-width of $A_{n}$ $(n\geq p)$ is at most $3$. This result is sharp, as there are families of alternating groups with $p$-width precisely 3, for each prime $p$.