A Generalization Of Huppert's Splitting Theorem And Its Applications

The problem concerning when a finite group is splitting over a normal subgroup is studied and the famous Huppert's splitting theorem of p-version is generalized to ?-version.A detailed proof is supplied here and a broader splitting criterion is obtained.As applications of the theorem,unified and simplified proofs of some classical theorems in the theory of transfer are given.The first main conclusion of this paper is as follows:Theorem 1.Let G be a group,H?<G and N(?)G.Write J = N ? H and suppose that K is a normal subgroup of H contained in J.Assume that(1)J/K is abelian,(2)| J:K| and |G:H| are coprime,(3)N ? A?(N),and ? = ?(J/K).Then J/K has a complement in H.In other words,' there is a subgroup X ? H such that JX = H and J ? X = K.Huppert's splitting theorem studies the existence of the complement of a normal abelian subgroup in the whole group.So we consider when an invariant normal a-belian subgroup has a stable complement under group actions.This is also a natural generalization of Huppert's splitting problem.The following is the second result of this paper,which generalizes the famous Maschke's theorem in group representation theory.In fact,by means of cohomology,we get an effective criterion.Theorem 2.Suppose that G is a group,acted on by a group A.Assume that N is an A-invariant normal abelian subgroup of G and write Q = G/N.If N has a complement in G,then we can uniquely define a cohomology element ??H1(A,Der(Q,N)),such that N has an A-invariant complement in G if and only if ?=0.Similarly,using the properties of cohomology,we can reduce the existence of the stable complement subgroup to the case that the operator group is a p-group.Theorem 3.Suppose that G is a group,acted on by a group A.Assume that N is an A-invariant normal abelian subgroup of G and that N has a complement in G.For every prime factor p of |A|,let Ap be a Sylow p-group of A.Then N has an A-in,variant complement in G if and only if N has an A-invariant complement in G for every p.