Frobenius' result on simple groups of order (p^3-p)/2

TL;DR

This paper discusses Frobenius' 1902 proof that PSL_2(Z/p) is the unique simple group of order (p^3-p)/2 for p>3, including a modern proof suitable for undergraduate courses.

Contribution

It presents a version of Frobenius' original argument and a modern proof of the simplicity of PSL_2(Z/p) for p>3, aimed at undergraduate students.

Findings

Frobenius proved the uniqueness of PSL_2(Z/p) as the simple group of that order.

A modern, accessible proof of PSL_2(Z/p) simplicity is provided.

The results are relevant for understanding classification of finite simple groups.

Abstract

The complete list of pairs of non-isomorphic finite simple groups having the same order is well-known. In particular for p>3, PSL_2(Z/p) is the "only" simple group of order (p^3-p)/2. It's less well-known that Frobenius proved this uniqueness result in 1902. This note presents a version of Frobenius' argument that might be used in an undergraduate honors algebra course. It also includes a short modern proof, aimed at the same audience, of the much earlier result that PSL_2(Z/p) is simple for p>3; a result stated by Galois in 1832.