A Characterization of the Finite Simple Groups,

In this paper, we obtain a quantitative characterization of all finite simple groups. Let πt(G) denote the set of indices of maximal subgroups of group G and let P(G) be the smallest number in πt(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 πt(N) πt(G), then • or • = and (2, 11) or = (2) and (3). Theorem 3 Let N be a finite simple and let G be a finite group. If G = N and πt(G) = πt(N), then G N.