Throughout the paper we consider only finite groups.
In [l] W. GASCHUTZ introduced the notion of formation which is useful and convenient for stu.dying not only groups but also other algebraic systems (see [2]). Recall that a class 3 of groups is called a formation if it is closed under taking homomorphic images and G / A E 3, G / B E 7 always implies CIA n B E 3. A very significant notion of local formation is contained in [l] as well. A non-empty formation 3 is called local if there exists a function f, assigning to every prime p a formation f ( p ) , such that If is the class of all groups G having a chief series G = GO 3 G I 3 . . . 3 Gt = 1 in which G/CG(Gi-l/Gi) E f(p) for every i and every prime p dividing IGi-1 : Gi(; ,hat function f is called a local screen of F. By the celebrated Gaschuta -Lubeseder -Schmid theorem [3; IV, 4.61, the following statements about a non-empty formation T are equivalent: a) 3 is local, b) F is saturated, i.e., G/$(G) E 3 always implies G E 3.
It is clear that the intersection of all local formations containing some class X of groups is again a local formation, which is the smallest local formation containing A!. This formation is denoted by lform(X). It is well known that the set of all local formations is a subsernigroup of the semigroup of all formations (the Gaschutz product FG of the formations 3 and 6 is the class of all groups whose G-residuals belong Let p be a prime. Following [4] we say that a non-empty formation 3 is p-local if Iform(F) is contained in NPt3, where p' is the set of all primes different from p and Npt is the class of all nilpotent p'-groups. In [5] it was proved that a non-empty formation 3 is p-local if and only if it is p-saturated, i.e., G/(O,(G) n +(G)) E 3 always implies G E F. This result does not follow from FORSTER'S generalization [6] of the Gaschutz -Lubeseder -Schmid theorem. It is clear that a non -empty formation is local if it is p-local for all p in the set P of all primes. 1.0 3). 1991 Maihernatio Subject Classificaiion. Primary 20D10. Keywords and phrases. Finite groups, formations. Ballester -Bolinches/Shemetkov, On Lattices of p -Local Formations of Finite Groups 59