A counterexample for the conjecture of finite simple groups

TL;DR

This paper presents counterexamples to a conjecture claiming finite simple groups can be uniquely identified by their order and the number of elements of order p, challenging a key assumption in group theory.

Contribution

It provides the first known counterexamples to the conjecture that finite simple groups are determined solely by their order and element counts of certain orders.

Findings

Counterexamples disprove the conjecture.

Finite simple groups cannot always be uniquely identified by order and element counts.

Challenges existing methods for classifying finite simple groups.

Abstract

In this note we provide some counterexamples for the conjectures of finite simple groups, one of the conjectures said "all finite simple groups $G$ can be determined using their orders $|G|$ and the number of elements of order $p$, where $p$ the largest prime divisor of $|G|$".