<html><head><title>[format] 4 FNormalizers</title></head> <body text="#000000" bgcolor="#ffffff"> [<a href = "chapters.htm">Up</a>] [<a href ="C003S000.htm">Previous</a>] [<a href ="C005S000.htm">Next</a>] [<a href = "theindex.htm">Index</a>] <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> [<a href = "chapters.htm">Up</a>] [<a href ="C003S000.htm">Previous</a>] [<a href ="C005S000.htm">Next</a>] [<a href = "theindex.htm">Index</a>] <P> <address>format manual<br>February 2003 </address></body></html>