Computing Formation Theoretic Subgroups and Certain Complements in F Inite Solvable Groups

In 1963 Gasch utz proved the existence in nite solvable groups of certain characteristic conjugacy classes of subgroups associated with formations. His initial work in 6] has developed into an extensive theory. In this paper we introduce a new approach to this theory and theoretical results that yield practical methods to compute formation theoretic subgroups such as F-normalizers and F-covering subgroups in nite solvable groups. Our methods are based on nding normalizers of the p-complements of selected normal subgroups avoiding direct use of chief factors. We also use our normalizer algorithm to construct certain types of complements in nite solvable groups, including the head complements associated with special polycyclic generating sequences and the LG-series used by C. R. Leedham-Green to develop special power commutator presentations of nite solvable groups. Since these presentations have proved to be of central importance for eecient computations with nite solvable groups (see 1]), a practical method to obtain them has important applications. Our algorithms are based on the concept of \exhibiting" subgroups of nite solvable groups. The idea of exhibiting subgroups has been introduced by C. R. Leedham-Green and is described in in 1] and 5]. The key to our approach will be a characterization of those sets of subgroups that can be simultaneously exhibited. The paper is organized as follows. First we describe the fundamental idea of exhibiting subgroups of nite solvable groups, and give its basic applications in Section 1. Then in Section 2 we introduce the key algorithm of our paper, a method to exhibit a certain type of normalizer in nite solv-able groups. In Section 3 we show how to use such normalizers to compute certain complements, and in Section 4 we describe their applications for the computation of formation theoretic subgroups. We report on the implementations of these methods in the computer algebra system GAP, see 10], and give runtimes in Section 5. Finally, in Section 6 we outline a complexity analysis.