Local formations in which every subformation of type N_P has a complement

摘要

ALL the groups considered in this note are finite. Definitions and notations are the same as those given in refs. Here we give some definitions and notations frequently used in this note. A class of groups is called a formation if it is closed undersubdirect product and ho-momorphic image. A nonempty formation F is called local provided that if G/Φ(G)∈F, then G∈F, where Φ(G) is the Frattini subgroups of G. A class of groups is called a Fitting class provided the following two conditions are satisfied: ( i ) if G∈Fand N△G, then N∈ F; ( ii ) if N_1, N_2, ..., N_t are subnormal subgroups of G and N_i∈F, i = 1, 2, ..., t, then ∈ F. If a local subformation of a formation is a Fitting class at the same time, then it is called a Fitting local subformation. Let X be a set of groups. We denote by form X the formation generated by X, by form X the local formation generated by X. π (G) denotes the set of prime factors of the order of G, π (F)=Uπ(G), N the formation of all nilpotent groups, N_π denotes the formation of all nilpotent π-groups. denotes the formation of unit element group. A subformation F_1 of formation F is said to have a complement in F, if there is a subformation F_2 of F such that F_1 ∩ F_2 = (1) and F=form (F_1 ∪ F_2).