If F is a locally defined integrated formation in GAP
and G is a finite solvable group, then the function FNormalizerWrtFormation
returns an F-normalizer of G. The function SystemNormalizer
yields a system normalizer of G.
The underlying algorithm here requires G to have a special pcgs (see SpecialPcgs), so the algorithm's first step is to compute such a pcgs for G if one is not known. The complement basis Sigma associated with this pcgs is then used to compute the F-normalizer of G with respect to Sigma. This process means that in the case of a finite solvable group G that does not have a special pcgs, the first call of FNormalizerWrtFormation (or similarly of FormationCoveringGroup) will take longer than subsequent calls, since it will include the computation of a special pcgs.
The FNormalizerWrtFormation algorithm next computes an F-system for G, a complicated record that includes a pcgs corresponding to a normal series of G whose factors are either F-central or F-hypereccentric. A subset of this pcgs then exhibits the F-normalizer of G determined by Sigma. The list ComputedFNormalizerWrtFormations( G ) stores the F-normalizers of G that have been found for various formations F.
The FNormalizerWrtFormation function can be used to study the subgroups of a single group G, as illustrated in an example in Section Other Applications. In that case it is sufficient to have a function ScreenOfFormation that returns a normal subgroup of G on each call.