On the theorems of Gaschütz and Willems (Q580489)

Let p be a prime, \(F=Z/(p)\) and G a finite group. A chief factor H/K of G is called complemented if H/K has a complement in G/K. Suppose \(E: \{\) \(1\}\) \(=G_ 0\subset G_ 1\subset...\subset G_ n=G\) is a fixed chief series of G and U is an irreducible FG module. Define \(m(E,U)=| \{i|\) \(G_ i/G_{i-1}\cong U\) and \(G_ i/G_{i-1}\) is complemented\(\}|\). Let J be the radical of FG and e be a primitive idempotent of FG such that eFG/eJ is isomorphic to the trivial FG module F. In this paper, the authors give an elementary proof of a theorem of W. Gaschütz [see \textit{B. Huppert} and \textit{N. Blackburn}, Finite Groups II (1982; Zbl 0477.20001), 15.5] which states: Suppose that G is a finite p solvable group. If U is a simple component of \(eJ/eJ^ 2\), then U must be isomorphic to a complemented p chief factor of G and it appears exactly m(E,U) times in \(eJ/eJ^ 2\). And also of a theorem of \textit{W. Willems} [Commun. Algebra 13, 2433-2447 (1985; Zbl 0575.20012)] which states: If V is a complemented chief factor of G, then it appears as a component of \(eJ/eJ^ 2\) with multiplicity at least m(E,V).