Finite simple non-Abelian groups (Q760498)

The main result of this article is: Theorem: Let \({\mathfrak M}\) be a class of finite simple non-Abelian groups up to isomorphism that contains all alternating groups \(A_ n\) for \(n\geq 5\). Assume also that whenever \({\mathfrak M}\) contains G and A is a simple non-Abelian subgroup of G such that: 1) \(N_ G(A)\) is a maximal subgroup of G and 2) \(C_ G(A)=1\) and 3) \(G=MA\) for every proper subgroup M of G of minimal index for which the permutation representation of A on the cosets of M is primitive, then \({\mathfrak M}\) also contains A. Under these conditions \({\mathfrak M}\) is the class of all finite simple non-Abelian groups.