Minimal generation of finite soluble groups by projectors and normalizers

Abstract

In this paper G denotes a non-identity finite soluble group. If A is an irreducible G-module, EndGA is a division ring by Schur's Lemma, actually a field, since G finite forces A to be finite. Moreover A is a vector space over EndGA with respect to . We let . Any chief factor of G is an irreducible G-module via the conjugation action, and it is central precisely when it is a trivial G-module. By a refined version of the Theorem of Jordan-Hölder [1, p. 33] the number of complemented chief factors of G, which are G-isomorphic to a given A, is constant for any chief series of G. We say that A is complemented, as aG-module, if .