Let F be an integrated locally defined formation, and let G be a finite solvable group with Sylow complement basis S: = { Sp | p divides ||G|| }. Let p be the set of prime divisors of the order of G that are in the support of F and [`(p)] the remaining prime divisors of the order of G. Then the F-normalizer of G with respect to S is defined to be Çp Î [`(p)] Sp ÇÇp Î p NG( GF(p) ÇSp ). The special case F(p) = { 1 } for all p defines the formation of nilpotent groups, whose F-normalizers Çp NG( Sp ) are the system normalizers of G. The F-normalizers of a group G for a given F are all conjugate. They cover F-central chief factors and avoid F-hypereccentric ones.
FNormalizerWrtFormation( G, F ) O
SystemNormalizer( G ) A
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
S associated with this pcgs is then used to compute the
F-normalizer of G with respect to S. 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
S. 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.
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