A counterexample to Herzog's Conjecture on the number of involutions

TL;DR

This paper provides a counterexample to Herzog's conjecture, which claimed that two simple groups with the same number of involutions must have the same order, challenging a longstanding assumption in group theory.

Contribution

The paper presents the first known counterexample disproving Herzog's conjecture about simple groups and involutions.

Findings

Counterexample disproves Herzog's conjecture

Same number of involutions does not imply same group order

Challenges previous assumptions in group theory

Abstract

In 1979, Herzog put forward the following conjecture: if two simple groups have the same number of involutions, then they are of the same order. We give a counterexample to this conjecture.