Historical note In 1831, Galois (1811–1832) was the first to comprehend that the algebraic solution of an equation was linked to the structure of a group of permutations associated with the equation. By 1832, Galois had identified the significance of special subgroups (now termed normal subgroups). He called the decomposition of a group into cosets of a subgroup a proper decomposition, if the right and left coset decompositions coincided. Galois later demonstrated that the non-Abelian simple group of the smallest order has an order of 60.