On some conjectures related to finite nonabelian simple groups

TL;DR

This paper presents counterexamples to Moretó's conjecture on finite simple groups, showing it fails generally but holds for sporadic and most alternating groups, and discusses related conjectures.

Contribution

It provides counterexamples to a conjecture about finite simple groups and verifies the conjecture for specific classes like sporadic and most alternating groups.

Findings

Counterexamples disprove the conjecture for some finite simple groups.

The conjecture holds for all sporadic simple groups.

The conjecture holds for alternating groups A_n, except for n=8,10.

Abstract

In this note we provide some counterexamples for the conjecture of Moret\'{o} on finite simple groups, which says that any finite simple group $G$ can determined in terms of its order $|G|$ and the number of elements of order $p$, where $p$ the largest prime divisor of $|G|$. Moreover, we show that this conjecture holds for all sporadic simple groups and alternating groups $A_n$, where $n\neq 8, 10$. Some related conjectures are also discussed.