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Galois and the simple group of order 60

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Abstract

In his testamentary letter to Auguste Chevalier, Évariste Galois states that, in modern terminology, the smallest simple group has order 60. No proof of this statement survives in his papers, and it has been suggested that a proof would have been impossible using the methods available at the time. We argue that this assertion is unduly pessimistic. Moreover, one fragmentary document, dismissed as a triviality and misunderstood, looks suspiciously like cryptic notes related to this result. We give an elementary proof of Galois’s statement, explain why it is likely that he would have been aware of the methods involved, and discuss the potential relevance of the fragment.

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Notes

  1. The term ‘general’ here indicates that the coefficients are arbitrary independent unknown quantities, rather than specific numbers. With specific numbers, some quintics are obviously soluble by radicals: for example \(x^5-2=0\) has the solution \(x=\root 5 \of {2}\).

  2. C’est aujourd’hui une vérité vulgaire que les équations générales de degré supérieure au \(4^{\textrm{e}}\) ne peuvent pas se résoudre par radicaux, c’est à dire que leurs racines ne peuvent s’exprimer par des fonctions des coëfficients qui ne contiendraient d’autres irrationnelles que des radicaux. Cette vérité est devenue vulgaire, en quelque sorte par ouï-dire et quoique la plupart des géomètres en ignorent les démonstrations présentées par Ruffini, Abel, &c. démonstrations fondées sur ce qu’une telle solution est déjà impossible au cinquième degré.

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  1. Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK

    Ian Stewart

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Stewart, I. Galois and the simple group of order 60. Arch. Hist. Exact Sci. 78, 1–28 (2024). https://doi.org/10.1007/s00407-023-00319-9

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