The Influence Of Conditional C-normal Subgroups On The Structure Of Finite Groupst

Let G be a finite group. A subgroup H is called conditional c-normal if there exists asubgroup N of G such that HN G and H∩N≤HG. In this paper, by using conditional cnormalityof some special subgroups(such as minimal subgroups, maximal subgroups, Sylowsubgroups, maximal subgroups of Sylow subgroups)of G, we obtain some sufficient or necessaryconditions for a finite group to be solvable, supersolvable, nilpotent. Some previouslyknown results are generalized.The thesis is divided into four parts according to contents.In part 1, the relationship between the c-normal subgroups and the conditional c-normalsubgroups was studied, and we give some definition and lemma. Using these lemmas, someknown theorems and properties about c-normal subgroups were generalized by use of conditionalc-normal subgroups.In part 2, by using the conditional c-normality of maximal subgroups, some sufficientor necessary conditions were obtained. In addition, we consider the relationship betweenthe normal index of maximal subgroups and the conditional c-normal subgroups. We obtainsome main results as follows:Theorem 2.1.1 Let G be a finite group. Then G is solvable if and only if everymaximal subgroup of G is conditional c-normal in G.Theorem 2.1.4 Let G be a finite group. Then G is solvable if and only if there existsa solvable conditional c-normal maximal subgroup M in G.Theorem 2.1.10 Let G be a finite group. Then G is solvable if and only if M isconditional c-normal in G for every maximal subgroup M in Fsc(G).In part 3, we use the conditional c-normality of maximal and minimal subgroups tocharacterize the structure of groups, obtain some sufficient conditions for supersolvablity ofa finite group. We obtain some main results as follows: Theorem 2.2.1 Let G be a finite group. If N is a normal subgroup of G such thatG/N is supersolvable and P1 is conditional c-normal in G for every Sylow subgroup P of Nand every maximal subgroup P1 of P. Then G is supersolvable.Theorem 2.2.7 Let F be a saturated formation containing U. Suppose that G is agroup with a solvable normal subgroup H such that G/H∈F. If all maximal subgroups ofall Sylow subgroups of F(H) are conditional c-normal in G, then G∈F.Theorem 2.2.14 Let G be a finite group with a normal subgroup H such that G/His supersolvable. If all maximal subgroups of any Sylow subgroup of F?(H) are conditionalc-normal in G, then G is supersolvable.In part 4, using conditional c-normality of some special subgroups, we obtain somesufficient conditions for a finite group to be nilpotent. We obtain some main results asfollows:Theorem 2.3.1 Let p be an odd prime dividing the order of a group G and P bea Sylow p-subgroup of G. If NG(P) is p-nilpotent and every maximal subgroup of P areconditional c-normal in G, then G is p-nilpotent.Theorem 2.3.5 Let p be a fixed prime. If every minimal p-subgroup of G is containedin Z(G), every cyclic subgroup of order 4 of G is conditional c-normal in G, then G isp-nilpotent.Theorem 2.3.9 Let p be a fixed prime. If every minimal p-subgroup of G is containedin Z_∞(G), every cyclic subgroup of order 4 of G is conditional c-normal in G, then G isp-nilpotent.