Some Conclusions About Finite Subsimple Groups

Suppose that G is always a finite group.This paper researched the subsimple group,which only includes a proper normal subgroup.The mian contents of this paper include some simple properties of subsimple groups,the sufficient conditions for the solvability and supersolvability of subsimple groups,some examples of subsimple groups, the structure of some special subsimple groups and the properties of characteristically subsimple groups and so on. The main methods of our research include group representation and group action, reversed proof, analysis etc.By studing the subsimple groups, I have gained these following main results:Theorem1 Let G be a subsimple group, N is its only proper normal subgroup, if G is nonperfect, N is abelian, then G is solvable.Theorem2 Let G be a subsimple group, N is its only proper normal subgroup, |N| = p, p is a prime, if G/N is solvable, then G is supersolvable.Theorem3 Let N be a normal subgroup of G, |G/N|=2, and N is simple group, G is non-decomposition group, then G is a subsimple group.Theorem4 |G| = 60, G is subsimple group,then the Sylow 5- subgroup of G is the only nontrivial normal subgroup of G.Theorem5 Let G be a finite group and let K be a field whose characteristic does not divide the order of G,φ₁,φ₂,…,φ_s are all inequivalent irreducible K-representations of G,φ₁ is main representation of G. Let N be only proper normal subgroup, if (?)i, we have kerφ_i = 1 or N, then G is subsimple.Theorem6 Let n≥5,n≠6, then the automorphism group of A_n is sub-simple group.