Minimal generation of finite solublegroups by projectors and normalizers

Abstract

In thispaper G denotes a non-identity finite soluble group. IfA is an irreducible G-module,End_GA is a division ring by Schur's Lemma,actually a field, since G finite forces A to befinite. Moreover A is a vector space overEnd_GA with respect to$\alphaa:=\alpha(a),\alpha\in\rm{End}_GA,a\inA$. We let$\varphi_G(A):=\rm{dim}_{\rm{End}_GA}A$. Any chief factor of Gis an irreducible G-module via the conjugation action, and itis central precisely when it is a trivial G-module. By arefined version of the Theorem of Jordan-Hölder [1, p. 33]the number $\delta_G(A)$ of complemented chief factors of G,which are G-isomorphic to a given A, is constant forany chief series of G. We say that A iscomplemented, as aG-module, if$\delta_G(A)>0$.