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<h1>4 FNormalizers</h1>

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.

<a name = ""></a>
<li><code>FNormalizerWrtFormation( </code><var>G</var><code>, </code><var>F</var><code> ) O</code>
<a name = ""></a>
<li><code>SystemNormalizer( </code><var>G</var><code> ) A</code>

If <var>F</var> is a locally defined integrated formation in GAP and
<var>G</var> is a finite solvable group, then the function <code>FNormalizerWrtFormation</code>
returns an <var>F</var>-normalizer of <var>G</var>. The function <code>SystemNormalizer</code> yields a
system normalizer of <var>G</var>.

The underlying algorithm here requires <var>G</var> to have a special pcgs (see section <a href="../../../doc/ref/chap45.html#X86007B0083F60470">Polycyclic Groups</a> in the GAP reference manual), so the algorithm's first step is
to compute such a pcgs for <var>G</var> if one is not known. The complement basis
Σ associated with this pcgs is then used to compute the
<var>F</var>-normalizer of <var>G</var> with respect to Σ. This process means that
in the case of a finite solvable group <var>G</var> that does not have a special pcgs,
the first call of <code>FNormalizerWrtFormation</code> (or similarly of <code>FormationCoveringGroup</code>)
will take longer than subsequent calls, since it will include the
computation of a special pcgs.

The <code>FNormalizerWrtFormation</code> algorithm next computes an <var>F</var>-system for <var>G</var>, a
complicated record that includes a pcgs corresponding to a normal series
of <var>G</var> whose factors are either <var>F</var>-central or <var>F</var>-hypereccentric. A subset
of this pcgs then exhibits the <var>F</var>-normalizer of <var>G</var> determined by
Σ. The list <code>ComputedFNormalizerWrtFormations( </code><var>G</var><code> )</code> stores the <var>F</var>-normalizers
of <var>G</var> that have been found for various formations <var>F</var>.

The <code>FNormalizerWrtFormation</code> function can be used to study the subgroups of a
single group <var>G</var>, as illustrated in an example in Section <a href="CHAP007.htm">Other Applications</a>. In that case it is sufficient to have a function
<code>ScreenOfFormation</code> that returns a normal subgroup of <var>G</var> on each call.


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