The Influence Of Conjugacy Classes Of Non-normal Subgroups And Promotion Of Normality On The Structure Of Finite Groups

In group theory, it is a wellknown fact that the properties of subgroups have a significant impact on the structure of group. It is a very important topic to discuss the structure of finite group through investigating the properties of the conjugates of some subgroups. There are the following three methods to study finite groups:(1) To study by using the number of the conjugates of some special subgroups. For example, Sylow Theorem.(2) To study by using the conjugacy classes of some special subgroups. For example, finite groups having exactly1,2,3or4conjugacy classes of non-normal subgroups are classified(see [1-4]).(3) To study by using the subgroup of enchangeable product. The properties of finite groups are obtained by some,s-c-permutable subgroups(H a subgroup of G is called s-c-permutable in G if there exists some x∈G such that HTx=TxH for all Sylow subgroup T of G) and some X-s-semipermutable subgroups(A is said to be X-s-semipermutable in G if A is X-permutable with every Sylow subgroup of some supplement T of A in G)(see [14] and [20]).In charter3, classifications of finite nilpotent groups having exactly1,2,3or4conjugacy classes of non-normal subgroups were completed, here we continue this work and classified finite nilpotent groups having exactly5conjugacy classes of non-normal subgroups.Theorem3.1Let G be a nilpotent group. Then v(G)=5if and only if G is isomorphic to one of the following groups.(1)M(pn)×Cq4, where p. q are distinct primes;(2)M(pn)×C2×C2, where p is an odd prime;(3)[Cs]C8;(4)<a,b.c|a8=1\,b4=c,c2=1, ab=a-1c,[a,c]=1>;(5)[C4]C8;(6)<a, b, c|a4=1,b16=c,c2=1,a6=a-1c,[a,c]=1>;(7)S32.In chapter4, we introduced the concept of weakly s-c-permutable subgroups.Definition4.1Let G be a finite group. A subgroup H of G is called weakly s-c-permutable in G if G has a subgroup T such that G=HT and H n T is s-c-permutable in G.We study the nilpotency or solvability of finite groups by weakly,s-c-permutable subgroups.Theorem4.3A group G is p-supersoluble if and only if there exists a p-soluble normal subgroup N of G such that G/N is p-supersoluble and every cyclic p-subgroup of N is weakly s-c-permutable in G.Theorem4.5Let (?) be a saturated formation containing (?) and G a group. Then G∈(?) if and only if there exists a soluble normal subgroup N of G such that G/N∈(?) and every cyclic subgroup of Ar with a prime power order is weakly s-c-pennutable in G.Theorem4.6Let (?) be a saturated formation containing il and G a group. Then G∈(?) if and only if there exists a soluble normal subgroup N of G such that G/N∈(?) and all maximal subgroups of all Sylow subgroups of F(N) are weakly s-c-permutable in G.In chapter5, we investigate further the influence of X-s-semipermutability of some subgroups on the structure of finite groups. Some new criteria for a group G to be p-nilpotent are obtained. Theorem5.1Let P be a Sylow p-subgroup of a group G, where p is the smallest prime dividing the order of G. Set X=Op’p(G). Then G is p-nilpotent if and only if every maximal subgroup of P not having a p-nilpotent supplement in G is X-s-semipermutable in G.Theorem5.2Let P be a Sylow p-subgroup of a group G, where p∈π(G). and let X=Op’p(G). Then G is p-nilpotent if and only if NG(P) is p-nilpotent and every maximal subgroup of P not having a p-nilpotent supplement in G is X-s-semipermutable in G.