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Generalized projectors and the saturated closure of a \(\pi\)-homomorph of finite \(\pi\)-solvable groups. (English) Zbl 1240.20016

Summary: The paper introduces and studies the notion of “generalized projector”, which generalizes the well-known notion of “projector” defined by W. Gaschütz [in Selected topics in the theory of soluble groups (Australian National University, Canberra) (1969)] as a generalization of the “covering subgroups” introduced by the same author [in Math. Z. 80, 300-305 (1963; Zbl 0111.24402)]. Let \(\pi\) be an arbitrary set of primes. A new definition for the “saturated closure” of a \(\pi\)-homomorph of finite \(\pi\)-solvable groups, equivalent to that of R. Covaci [Mathematica 35, No. 2, 137-139 (1993; Zbl 0832.20033)], is given. A property connected with the notion of generalized projector on a class \(X\) of finite \(\pi\)-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 \(\pi\)-homomorphs of finite \(\pi\)-solvable groups with the GP-property by means of the generalized projectors; 2) a theorem showing that if \(X\) is a \(\pi\)-homomorph of finite \(\pi\)-solvable groups with the GP-property and \(\overline X\) is its saturated closure, then \(X\) is a Schunck class if and only if \(X=\overline X\). These results prove that theorems similar to those obtained by J. Weidner [in Bull. Lond. Math. Soc. 8, 38-40 (1976; Zbl 0344.20015)] for finite solvable groups can be also obtained in the more general case of finite \(\pi\)-solvable groups.

MSC:

20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
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