Sophie: gap-system-4.4.12-5mdv2010.0 i586

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<html><head><title>[format] 4 FNormalizers</title></head>
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<h1>4 FNormalizers</h1><p>
<p>
Let <var><font face="helvetica,arial">F</font></var> be an integrated locally defined formation, and let <var>G</var> be 
a finite solvable group with Sylow complement basis <var>Sigma:= 
{ S<sup>p</sup> midp</var> divides  <var> |G| }</var>.  Let <var>pi</var> be the set of prime
divisors of the order of <var>G</var> that are in the support of <var><font face="helvetica,arial">F</font></var> and 
<var>overlinepi</var> the remaining prime divisors of the order  of <var>G</var>. 
Then the <strong><var><font face="helvetica,arial">F</font></var>-normalizer</strong> of <var>G</var> with respect to <var>Sigma</var> is defined 
to be  
<var>bigcap<sub>p inoverlinepi</sub> S<sup>p</sup> cap
 bigcap<sub>p inpi</sub> N<sub>G</sub>( G<sup><font face="helvetica,arial">F</font>(p)</sup> capS<sup>p</sup> )</var>. 
The special case <var><font face="helvetica,arial">F</font>(p) = { 1 }</var> for all <var>p</var> defines the formation 
of nilpotent groups, whose <var><font face="helvetica,arial">F</font></var>-normalizers <var> bigcap<sub>p</sub> N<sub>G</sub>( S<sup>p</sup> )</var> 
are the <strong>system normalizers</strong> of <var>G</var>. The <var><font face="helvetica,arial">F</font></var>-normalizers of a group 
<var>G</var> for a given <var><font face="helvetica,arial">F</font></var> are all conjugate. They cover <var><font face="helvetica,arial">F</font></var>-central chief 
factors and avoid <var><font face="helvetica,arial">F</font></var>-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 SpecialPcgs), so the algorithm's first step is
 to compute such a pcgs for <var>G</var> if one is not known. The complement basis
<var>Sigma</var> associated with this pcgs is then used to compute the
<var>F</var>-normalizer of <var>G</var> with respect to <var>Sigma</var>. 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
<var>Sigma</var>. 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="C007S000.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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<P>
<address>format manual<br>February 2003
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