Symmetric Generation of $J_2$ on 32 Letters

TL;DR

This paper provides a computer-free proof demonstrating that the group $J_2$ is isomorphic to a specific symmetric generator-based construction involving 32 letters, with explicit relations of lengths 3 and 6.

Contribution

It introduces a novel, computer-free proof of the isomorphism of $J_2$ using symmetric generators and explicit relations, simplifying previous computational approaches.

Findings

$J_2$ is isomorphic to a symmetric generator-based progenitor

Explicit relations of lengths 3 and 6 define the isomorphism

Proof avoids computational methods, using only algebraic arguments

Abstract

We give a computer-free proof that $J_2$ is isomorphic to the progenitor $2^{\star 32}: (2^{1+4}:A_5)$ factored by two relations, one of length 3 and and one of length 6, in the symmetric generators.