Covaci, Rodica Generalized projectors and the saturated closure of a \(\pi\)-homomorph of finite \(\pi\)-solvable groups. (English) Zbl 1240.20016 Stud. Univ. Babeș-Bolyai, Math. 56, No. 1, 3-13 (2011). 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 Keywords:finite groups; Schunck classes; homomorphs; projectors; saturated closures; \(\pi\)-solvable groups Citations:Zbl 0111.24402; Zbl 0832.20033; Zbl 0344.20015 PDFBibTeX XMLCite \textit{R. Covaci}, Stud. Univ. Babeș-Bolyai, Math. 56, No. 1, 3--13 (2011; Zbl 1240.20016)