Generating alternating and symmetric groups with two elements of fixed order

TL;DR

This paper investigates the conditions under which alternating and symmetric groups can be generated by two elements of a fixed order, providing a nearly complete characterization with some explicit exceptions.

Contribution

It answers a previously open question by establishing that most alternating and symmetric groups can be generated by two elements of the same order, with specific known exceptions.

Findings

Most $A_n$ and $S_n$ can be generated by two elements of order $k$

The paper identifies explicit exceptions to this generation property

Provides a comprehensive answer to Lanier's question

Abstract

We answer a question raised by Lanier about the possibility of generating $A_n$ and $S_n$ with two elements of order $k$, where $n \geqslant k \geqslant 3$. We show that this can always be done apart from some clear exceptions.