Computing Projectors, Injectors, Residuals and Radicals of Finite Soluble Groups

Let G be a finite soluble group, given by a polycyclic generating system, and X a class of groups, represented by an algorithm that decides whether a given finite group belongs to X or not. This paper contains practical algorithms for the computation of X -projectors and X -injectors of G,whereX is an arbitrary Schunck class or Fitting class, respectively. It also describes algorithms for computing X -radicals for Fitting classes X and X -residuals for formations X. Variants of the latter algorithms can be used to compute all normal (X -)subgroups and the socle of G.