Über die Längen von p-Reihen endlicher Gruppe. (On the lengths of p- series of finite groups) (Q1082438)

A p-series for a finite group G is a series \(\{N_ i|\) \(0\leq i\leq r\}\) of normal subgroups \(\{l\}=N_ r\subset...\subset N_ 1\subset N_ 0=G\) for which (i) \(p| | N_ i/N_{i+1}|\) for \(i=0,...,r-1\), (ii) \(O^ p(N_ i)=N_ i\) for \(i=0,...,r-1\), and (iii) no proper refinement of the series satisfies either (i) or (ii). Let \(m_ p(G)\) and \(l_ p(G)\) denote the maximum and minimum of the lengths of the p-series for G. Results generally associated with p-solvable groups and p-lengths are extended to arbitrary finite groups with respect to the p-series. On the other hand this approach lends itself to strengthening known results on p-solvable groups. The author's main theorem is the following: Let G be a finite p-solvable group of p-length \(r=l_ p(G)\) and k be a field of characteristic p. Then there are (r-1) complemented p-chief factors \(T/\bar T\) of G such that no irreducible \(k[G]\)-factor module of \(T/\bar T\otimes_{F_ p}k\) would be centralized by a Sylowp-subgroup of G. Different p-chief factors belong to nonisomorphic irreducible k[G]-factor modules. In the second section of this article, the author extends to p-series results related to the length of the series and the \(\Gamma\)-composition series of \(S/\Phi\) (S) for which \(\Gamma\subseteq Aut(G)\) and fixes the p-series as well as the Sylow p-subgroup S. For the special case of \(\Gamma\) being the identity map and G a p-solvable group, the well-known bounds on \(m_ p(G)\) are obtained.