The $(2,p)$-generation of sporadic simple groups

TL;DR

This paper proves that most sporadic simple groups can be generated by an involution and an element of prime order p, with only four exceptions for p=3, highlighting a specific generation property of these groups.

Contribution

It establishes a nearly universal generation property for sporadic simple groups involving involutions and elements of order p, except for four specific cases.

Findings

Most sporadic simple groups are generated by an involution and an element of order p.

Four sporadic groups are exceptions for p=3.

The result applies to all odd primes dividing the group order.

Abstract

In this short note we prove that, if $p$ is an odd prime dividing the order of a sporadic simple group, then with the exception of four groups for $p=3$, all sporadic simple groups are generated by an involution and an element of order $p$.