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# Frobenius’ Result on Simple Groups of Order $\frac{p^3 - p}{2}$

Paul Monsky
The American Mathematical Monthly
Vol. 120, No. 8 (October 2013), pp. 725-732
DOI: 10.4169/amer.math.monthly.120.08.725
Stable URL: http://www.jstor.org/stable/10.4169/amer.math.monthly.120.08.725
Page Count: 8
Item Type
Article
References
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## Abstract

Abstract The complete list of pairs of non-isomorphic finite simple groups having the same order is well known. In particular, for p > 3, PSL2(ℤ/p) is the “only” simple group of order \documentclass{article} \pagestyle{empty}\begin{document} $\frac{p^3 - p}{2}.$ \end{document} 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 PSL2(ℤ/p) is simple for p > 3, a result stated by Galois in 1832.