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4 FNormalizers

Let F be an integrated locally defined formation, and let G be a finite solvable group with Sylow complement basis \$\Sigma\$. Let π be the set of prime divisors of the order of G that are in the support of F and ν the remaining prime divisors of the order of G. Then the F-normalizer of G with respect to Σ is defined to be [see the PDF manual]. The special case F(p) = { 1 } for all p defines the formation of nilpotent groups, whose F-normalizers 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 section Polycyclic Groups in the GAP reference manual), so the algorithm's first step is to compute such a pcgs for G if one is not known. The complement basis Σ associated with this pcgs is then used to compute the F-normalizer of G with respect to Σ. 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 Σ. 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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FORMAT manual
March 2018