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RESEARCH PRODUCT

A Question of R. Maier Concerning Formations

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The formation f is said to be saturated if the group G belongs to f Ž . whenever the Frattini factor group GrF G is in f. Let P be the set of all prime numbers. A formation function is a Ž . function f defined on P such that f p is a, possibly empty, formation. A formation f is said to be a local formation if there exists a formation Ž function f such that f s G g G : if HrK is a chief factor of G and p < < Ž . Ž .. divides HrK , then GrC HrK g f p ; G is the class of all finite G groups. If f is a local formation defined by a formation function f , then Ž . we denote f s LF f and f is a local definition of f. Among all possible local definitions of a local formation f there exists exactly one, denoted by Ž Ž . . F, such that F is integrated i.e., F p : f for all p g P and full Ž Ž . Ž . . S F p s F p for all p g P; here S denotes the class of all p-groups ; p p this F is the canonical local definition of f. The celebrated w x Gaschutz]Lubeseder]Schmid Theorem 2, IV, Theorem 4.6 states that a formation is saturated if and only if it is local. Let f be a formation. It is known that each group G has a smallest normal subgroup with quotient in f; it is called the f-residual of G and is denoted by G. On the other hand, it is well known that formations are