A Symmetric Presentation for J₁

Abstract

We give a computer-free proof that the sporadic simple group J ₁ is a isomorphic to the progenitor 2*⁵ : A ₅ factorized over a single relation. Precisely, we prove that J ₁ is defined by the presentation ⟨x, y, t ∣ x ⁵ = y ³ = (xy)² = 1 = t ² = [y, t] = [y, t ^{x 3} ] = (xt)⁷⟩.

Key Words:

• Sporadic groups

Acknowledgments

This paper was written while the author was staying at Bar Ilan University. The research was supported in part by the Israel Science Foundation, founded by the Israel Academy for Sciences and Humanities. I thank all the members of the Department of Mathematics of Bar Ilan for their hospitality and, especially, Ron Adin and Yuval Roichman for their invitation.