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ABSTRACT. The paper introduces and studies the notion of generalized projector, which generalizes the well-known notion of projector defined by W. Gaschutz in [8] as a generalization of the covering subgroups introduced by the same author in [7]. Let π be an arbitrary set of primes.A new definition for the saturated closure of a π-homomorph of finite π- solvable groups, equivalent to that in [3], is given. A property connected with the notion of generalized projector on a class X of finite π-solvable groups, called the GP-property, is also introduced. The main results of the paper are the following: 1) a characterization theorem for the saturated closure of the π-homomorphs of finite π-solvable groups with the GP-property by means of the generalized projectors; 2) a theorem showing that if X is a π-homomorph of finite π-solvable groups with the GP-property and X is its saturated closure, then X is a Schunck class if and only if X = X. These results prove that theorems similar to those obtained by J. Weidner in [10] for finite solvable groups can be also obtained in the more general case of finite π-solvable groups.Mathematics Subject Classification (2010): 20D10.

Keywords: Schunck class, homomorph, projector, saturated closure of a homomorph, π-solvable group.