A Characterization of the Finite Simple Groups

Abstract

In this paper, we obtain a quantitative characterization of all finite simple groups. Let π t G denote the set of indices of maximal subgroups of group G and let P G be the smallest number in π t G . We have the following theorems.Theorem 2. Let N and G be finite simple groups. If N divides G , P N = P G , and π t N ⊆ π t G , then(2) G = M 11 and N ∼ = PSL 2 11 or G = S 6 2 and N ∼ = U 3 3 . Theorem 3. Let N be a finite simple and let G be a finite group. If G = N and π t G = π t N , then G ∼ = N. 