【摘要】 设群G为有限群,F(G)为其Fitting子群。若G/F(G)为幂零群,则称G为亚幂零群或MN-群。全部有限的MN-群类记为H 。本文所讨论的群假定为有限群。更多还原

【Abstract】 A finite group G is called an MN-group ,if G/F(G) is nilpotent. where F(G) is the Fitting subgroup of G. In this paper all groups considered will be finite.The main result is as follows:Theorem. If G is a finite non-MN-group, each of whose proper subgroupis an MN-group and φ(G) = 1 (φ(G) is the Frattini subgroup of G), then G is eithes a non-Abelian simple group or a solvable group. When G is a non-Abelian simple group, G is one of the following four types of simple groups.(i) PSL(2,p),where p is a prime with p<3,p2?1(mod 5) and p2?1(mod 16);(ii) PSL(2,2q) where q is a prime;(iii) PSL(2,3q) , where q is an odd prime;(iv) The Suzuki group Sz(2q), where q is an odd prime. When G is solvable, the structure of G is as follows: G=F(G)Mwhere F(G)∩M=1, CG(F(G))=F(G)and F(G)is the unique minimal normal subgroup of G and hence is an elementary Abelian p-group (p is a prime). M is a maximal subgroup of G and M is a minimal non-nilpotent group. M acts faithfully and irreducibly on F(G). Finally, M is one of the following two types:(i) r-elementary groups of order qβrγ, where q arid r are primes, and p, g and r are pairwise distinct, β and γ are positive integers.(ii) q-elementary groups of order pqβ, where q is a prime with q≠p, and β is a positive integer.更多还原