Let `F` be an integrated locally defined formation, and let `G` be
a finite solvable group with Sylow complement basis
$\Sigma$.
Let `pi` be the set of prime
divisors of the order of `G` that are in the support of `F` and
`nu` the remaining prime divisors of the order of `G`.
Then the ** F-normalizer** of

`FNormalizerWrtFormation( `

`, `

` ) O`

`SystemNormalizer( `

` ) 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
`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 2011