关于有限群两个超可解子群之积的问题

Products of Two Supersolvable Subgroups of a Finite Group

定义1 设H是G的子群,如果H同G的任一子群可交换。称H在G中拟正规。定义2 设H,K为G的子群,如果H同K中任一子群可交换,称H在K中拟正规我们知道两个正规子群之积不一定为超可解子群.在[1]中,Baer证明了,如果G为两个正规超可解子群之积,且G’幂零,则G为超可解群;在[2]中Friesen证明了两个指数互质的正规的超可解子群之积仍为超可解群;在[3]中Kegel证明了如果G=HK=HL=KL,而H,K为幂零子群,L为超可解子群,则G为超可解子群;

In this paper, we give some sufficient conditions for products of two supersolvable sub-groups to be supersolvable groups. Our results generalize some known results.Theorem 1 Let G = HK,(\H\,\K\) = 1, Where H and K are two supersolvable sub-groups. If H is commutative with every maximal subgroup of K, and K is commutative with every maximal subgroup of H, then G is supersolvable.Theorem 2 Let G = HK, H ∩ K = 1, H G, and K be quasinormal in H. If H, K are supersolvable, the G is supersolvable.Theorem 3 Let G = HK,(\H ,\K\) - l,H,K be two supersolvable subgroups. If H is commutative with any Sylow subgroup of K and any maximal subgroup of every sylow subgroup of K, and K is commutative with any sylow subgroup of H and any maximal subgroup of every sylow subgroup of H, then G is supersolvable.Theorem 4 If H, K are two supersolvable subgroups ofG,G= HK,G' is nilpotent, II is quasi normal in K, and K is quasi normal in H, then G is supersolvable.Theorem 5 If H, K are two supersolvable subgroups ofG,G = HK,H' G, [H, K] G, [H, K] is nilpotent, H is quasi normal in K, and K is quasi normal in H, then G is supersolvable.