Topic summary
Simple groups
Extracted from the Wikipedia article Simple group.
Construction
Simple groups have been studied at least since early Galois theory, where Évariste Galois realized that the fact that the alternating groups on five or more points are simple (and hence not solvable), which he proved in 1831, was the reason that one could not solve the quintic in radicals. Galois also constructed the projective special linear group of a plane over a prime finite field, PSL(2,p), and remarked that they were simple for p not 2 or 3. This is contained in his last letter to Chevalier, and are the next example of finite simple groups.