GASCHUTZ PRODUCTS OF GROUP CLASSES

All groups considered are finite. The smallest, by inclusion, normal s ubgroup K of a group G for which G/K is an element of h is called a h- residual normal subgroup of G. A Gaschutz product of two group classes M and h is the class Mh of all groups G such that K is an element of M for some h-residual normal subgroup K of G. Let n(X) denote the set of primes dividing the orders of groups in X. We set X/O-p' = {G/O-p' (G) : G is an element of X} if p is an element of pi(X), and X/O-p' = empty set if p is not an element of pi(X). We say that a group class X is p-local in a group class M if N-p(X/O-p') subset of or equal to M. A class X is called p-local if it is p-local in X. In particular, if a formation F is p-local for every prime p then it is local in sense o f Gaschutz. In the paper it is proved the following. Theorem. Let p be a prime and F = Mh a formation where M is a class of groups and h is a formation. Let H-h denote the set of h-residuals of all groups in F. Then the following assertions hold: a) if p is not an element of pi(F -h) then F is p-local if and only if h is p-local; b) if p is an eleme nt of pi(F-h) and the class F-h is p-local then the class F-h is p-loc al; c) if p is an element of pi(F-h) and F is p-local then the class F -h is p-local in the formation generated by F-h. Five open questions a re posed.