# 4 FNormalizers

Let F be an integrated locally defined formation, and let G be a finite solvable group with Sylow complement basis Sigma := { Sp mid p divides |G| }. Let pi be the set of prime divisors of the order of G that are in the support of F and overlinepi the remaining prime divisors of the order of G. Then the F-normalizer of G with respect to Sigma is defined to be bigcapp in overlinepi Sp cap bigcapp in pi NG( GF(p) cap Sp ). The special case F(p) = { 1 } for all p defines the formation of nilpotent groups, whose F-normalizers bigcapp 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 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.

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November 2000