<html><head><title>[FORMAT] 4 FNormalizers</title></head>
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<h1>4 FNormalizers</h1><p>
<p>
Let <b>F</b> be an integrated locally defined formation, and let <i>G</i> be
a finite solvable group with Sylow complement basis
$\Sigma$.
Let π be the set of prime
divisors of the order of <i>G</i> that are in the support of <b>F</b> and
ν the remaining prime divisors of the order of <i>G</i>.
Then the <strong><b>F</b>-normalizer</strong> of <i>G</i> with respect to Σ is defined
to be
[see the PDF manual].
The special case <b>F</b>(<i>p</i>) = { 1 } for all <i>p</i> defines the formation
of nilpotent groups, whose <b>F</b>-normalizers
are the <strong>system normalizers</strong> of <i>G</i>. The <b>F</b>-normalizers of a group
<i>G</i> for a given <b>F</b> are all conjugate. They cover <b>F</b>-central chief
factors and avoid <b>F</b>-hypereccentric ones.
<p>
<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>
<p>
If <var>F</var> is a locally defined integrated formation in <font face="Gill Sans,Helvetica,Arial">GAP</font> 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>.
<p>
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 <font face="Gill Sans,Helvetica,Arial">GAP</font> 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.
<p>
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>.
<p>
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.
<p>
<p>
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