仅含两个非次正规子群共轭类的有限群

Groups Which Have Only One Non-subnormal Subgrouops Conjugate Class

主要证明了:若有限群G只含两个非次正规子群共轭类H=(H_1,H_2,…,H_m)和K={K_1,K_2,…,K_n},则G可解.其中|G|含两个或三个素因子,且G满足下列情形之一:(1)G=H■Q,其中H是具有循环极大子群的p-群,Q是Sylow q-子群,p,q为互不相同的素数;(2)G=K■Q,其中K是G的循环Sylow p-子群,Q是G的Sylow q-子群;(3)G=A■B,其中A是p^mq^n阶非幂零有限内-Abel群,B是Sylow r-子群,p,q,r为互不相同的素数.

In this paper the authors mainly proved that： If the finite group G has two conjugate classes of non-subnormal subgroups H= {H1 ,H2 …,Hm} and K= {K1 ,K2 …,Kn} , then G is soluble, and ｜G｜ has at most three prime factors, and G satisting one of the following conditions： （1） G= H Q, where H is a p-group which has cyclic maximal subgroups, and Q is the Sylowq-subgroup of G, p and q are different primes. （2） G：K Q, where K is a Sylow p-subgroup of G, K is cyclic, and Q is the Sylow q-subgroup of G. （3） G：A B, where A is a non-nilpotent finite inner-abel group with order p^mq^n, B the a Sylow r-of G, and p, q, r are different primes.